Maxima CAS an introduction

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PROGRAMMING
MAXIMA
Arun Umrao
www.sdmsacademy.in
DRAFT COPY- GPL LICENSING

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Contents
I Modelica Commands 17
1 Maxima Core 19
1.1 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.1 Factorial Symbol (!) . . . . . . . . . . . . . . . . . . . . . 19
1.1.2 Double Factorial Symbol (!!) . . . . . . . . . . . . . . . . 19
1.1.3 Termination of Line (;) . . . . . . . . . . . . . . . . . . . 20
1.1.4 Negation of Expression (#) . . . . . . . . . . . . . . . . . 20
1.1.5 Evaluate Internally & No Output ($) . . . . . . . . . . . . 20
1.1.6 Do Not Execute Function (’) . . . . . . . . . . . . . . . . 20
1.1.7 Re-evaluation of Speciﬁc Input (“) . . . . . . . . . . . . . 21
1.1.8 Variable Assigning Operator (:) . . . . . . . . . . . . . . . 21
1.1.9 Operator Assigner (::) . . . . . . . . . . . . . . . . . . . . 21
1.1.10 Macro Function Deﬁnition Operator (::=) . . . . . . . . . 22
1.1.11 Function Operator (:=) . . . . . . . . . . . . . . . . . . . 22
1.1.12 Sum Operator (+) . . . . . . . . . . . . . . . . . . . . . . 23
1.1.13 Subtraction Operator (-) . . . . . . . . . . . . . . . . . . . 23
1.1.14 Product Operator (*) . . . . . . . . . . . . . . . . . . . . 24
1.1.15 Division Operator (/) . . . . . . . . . . . . . . . . . . . . 24
1.1.16 Exponential Operator (ˆ) . . . . . . . . . . . . . . . . . . 25
1.1.17 Equal Operator (=) . . . . . . . . . . . . . . . . . . . . . 26
1.1.18 Less Than Operator (<) . . . . . . . . . . . . . . . . . . . 26
1.1.19 Grater Than Operator (>) . . . . . . . . . . . . . . . . . 26
1.1.20 Less Than or Equals To Operator (<=) . . . . . . . . . . 27
1.1.21 Greater Than or Equals To Operator (>=) . . . . . . . . 27
1.1.22 Lisp Name (?) . . . . . . . . . . . . . . . . . . . . . . . . 27
1.1.23 A List ([ ... ]) . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.1.24 Call Last Garbage (%) . . . . . . . . . . . . . . . . . . . . 28
1.1.25 Previous Statement (%%) . . . . . . . . . . . . . . . . . . 28
1.1.26 Exponential (%e) . . . . . . . . . . . . . . . . . . . . . . . 28
1.1.27 %e tonumlog . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.1.28 Negative Exponent as a Quotient (%edispﬂag) . . . . . . 29
1.1.29 Demoivre Mode (%emode) . . . . . . . . . . . . . . . . . 29
1.1.30 Exponential Numerical (%enumer) . . . . . . . . . . . . . 29
1.1.31 Gamma Constant (%gamma) . . . . . . . . . . . . . . . . 30
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4 CONTENTS
1.1.32 Pi Value (%pi) . . . . . . . . . . . . . . . . . . . . . . . . 30
1.1.33 Previous Output (%th) . . . . . . . . . . . . . . . . . . . 30
1.1.34 Most Recent Expression ( ) . . . . . . . . . . . . . . . . . 30
1.1.35 Display String (disp) . . . . . . . . . . . . . . . . . . . . . 30
1.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.1 Error (error) . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.2 Error Size (errorsize) . . . . . . . . . . . . . . . . . . . . 31
1.2.3 Error Symbol (errorsyms) . . . . . . . . . . . . . . . . . 31
1.2.4 Error Message (errormsg) . . . . . . . . . . . . . . . . . . 31
1.2.5 Debug Mode Maxima (debugmode) . . . . . . . . . . . . 32
1.2.6 Show Examples (example) . . . . . . . . . . . . . . . . . . 32
1.2.7 Set Numerical Precision (fpprec) . . . . . . . . . . . . . . 32
1.2.8 Toggle to Floating Point (numer) . . . . . . . . . . . . . . 32
1.3 Procedural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.3.1 Interrupt Maxima (kill) . . . . . . . . . . . . . . . . . . . 33
1.3.2 Modes of Variables (modedeclare) . . . . . . . . . . . . . 33
1.3.3 Evaluate & Print Expression (print) . . . . . . . . . . . . 34
1.3.4 Labels (labels) . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3.5 Line Number (linenum) . . . . . . . . . . . . . . . . . . . 34
1.3.6 List of Options (myoptions) . . . . . . . . . . . . . . . . . 34
1.3.7 Ordinary Base (obase) . . . . . . . . . . . . . . . . . . . . 34
1.3.8 Print the Reset Option (optionset) . . . . . . . . . . . . . 35
1.3.9 Play Previous Garbage (playback) . . . . . . . . . . . . . 35
1.3.10 Prompting Symbol (prompt) . . . . . . . . . . . . . . . . 35
1.3.11 Quit the Session (quit) . . . . . . . . . . . . . . . . . . . . 35
1.3.12 Rationalized Print (ratprint) . . . . . . . . . . . . . . . . 35
1.3.13 Enter into Lisp System (tolisp) . . . . . . . . . . . . . . . 35
1.3.14 Call Functions of Current Session (backtrace) . . . . . . . 36
1.3.15 Break Maxima Evaluation (break) . . . . . . . . . . . . . 36
1.3.16 Bug Reporting Info (bugreport) . . . . . . . . . . . . . . 36
1.3.17 Build Information (buildinfo) . . . . . . . . . . . . . . . . 36
1.3.18 Debug Mode (debugmode) . . . . . . . . . . . . . . . . . 36
1.3.19 Show Evaluation Time (showtime) . . . . . . . . . . . . . 36
1.3.20 Evaluation Time (time) . . . . . . . . . . . . . . . . . . . 36
1.3.21 Promp & Read (read) . . . . . . . . . . . . . . . . . . . . 37
1.3.22 testsuiteﬁles . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.23 Test The Maxim (runtestsuite) . . . . . . . . . . . . . . . 37
1.3.24 Read Function Without Evaluation (readonly) . . . . . . 37
1.3.25 Activate Contexts (activate) . . . . . . . . . . . . . . . . . 38
1.3.26 List of Activated Contexts (activecontexts) . . . . . . . . 38
1.3.27 Additive Function (additive) . . . . . . . . . . . . . . . . 38
1.3.28 Alternative Name for Function (alias) . . . . . . . . . . . 38
1.3.29 Declare String as Alphabic (alphabetic) . . . . . . . . . . 38
1.3.30 Arguments In Expression (args) . . . . . . . . . . . . . . . 38
1.3.31 Atomic Expression (atom) . . . . . . . . . . . . . . . . . . 38
1.3.32 Declare a Function Antisymmetric (antisymmetric) . . . . 39

CONTENTS 5
1.3.33 Check The Integer Type (askinteger) . . . . . . . . . . . . 39
1.3.34 Retrive User Property (get) . . . . . . . . . . . . . . . . . 39
1.3.35 Declare Variable as Integer (integer) . . . . . . . . . . . . 39
1.3.36 Declare Variable as Non-Integer (noninteger) . . . . . . . 40
1.3.37 Integer Variabe (integervalued) . . . . . . . . . . . . . . . 40
1.3.38 Query for Sign of Expression (asksign) . . . . . . . . . . . 40
1.3.39 Declare Variables (assume) . . . . . . . . . . . . . . . . . 40
1.3.40 Declare Variable as Positive (assumepos) . . . . . . . . . 41
1.3.41 Declare As Communative Function (commutative) . . . . 41
1.3.42 Compare Expression (compare) . . . . . . . . . . . . . . . 41
1.3.43 Context (context) . . . . . . . . . . . . . . . . . . . . . . 41
1.3.44 Contexts (contexts) . . . . . . . . . . . . . . . . . . . . . 41
1.3.45 Deactivate Context (deactivate) . . . . . . . . . . . . . . . 41
1.3.46 Delcaration Statement (declare) . . . . . . . . . . . . . . 41
1.4 Declarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.4.1 Declare As Decreasing Function (decreasing) . . . . . . . 42
1.4.2 Declare As Increasing Function (increasing) . . . . . . . . 42
1.4.3 Declare Variable as Even Variable (even) . . . . . . . . . 42
1.4.4 Declare Variable as Odd Variable (odd) . . . . . . . . . . 42
1.4.5 Variable As a Constant (constant) . . . . . . . . . . . . . 42
1.4.6 Check a Constant (constantp) . . . . . . . . . . . . . . . . 42
1.4.7 Get Features (featurep) . . . . . . . . . . . . . . . . . . . 43
1.4.8 Set Feature (features) . . . . . . . . . . . . . . . . . . . . 43
1.4.9 Status (is) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.4.10 Kill Contextx (killcontext) . . . . . . . . . . . . . . . . . 43
1.4.11 Simplify as Left-Associative (lassociative) . . . . . . . . . 43
1.4.12 Recognize Function as N-Array (nary) . . . . . . . . . . . 44
1.4.13 Maximum Negative Expansion (maxnegex) . . . . . . . . 44
1.4.14 Match Declaration (matchdeclare) . . . . . . . . . . . . . 44
1.4.15 Simplify as Multiplicative (multiplicative) . . . . . . . . . 44
1.4.16 Multiply Two Expression (multthru) . . . . . . . . . . . . 44
1.4.17 Negative Distribution (negdistrib) . . . . . . . . . . . . . 44
1.4.18 Create New Context (newcontext) . . . . . . . . . . . . . 45
1.4.19 Numerical Value of Variable (numerval) . . . . . . . . . . 45
1.4.20 Pull Out Constant Term (outative) . . . . . . . . . . . . . 45
1.4.21 Positive Function (posfun) . . . . . . . . . . . . . . . . . . 45
1.4.22 Properties of Atom (properties) . . . . . . . . . . . . . . . 45
1.4.23 Assign a Value to Property (put) . . . . . . . . . . . . . . 46
1.4.24 Simplify as Right-Associative (rassociative) . . . . . . . . 46
1.4.25 Declare Variable as Irrational Variable (irrational) . . . . 46
1.4.26 Declare Variable as Irrational Variable (rational) . . . . . 46
1.4.27 Declare Variable as Complex Variable (complex) . . . . . 46
1.4.28 Declare Variable as Imaginary Variable (imaginary) . . . 46
1.4.29 Declare Variable as Real Variable (real) . . . . . . . . . . 47
1.4.30 Remove Indicator From Atom (rem) . . . . . . . . . . . . 47
1.4.31 Remove Property (remove) . . . . . . . . . . . . . . . . . 47

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1.4.32 Scalar Value (scalar) . . . . . . . . . . . . . . . . . . . . . 47
1.4.33 Sign of Expression (sign) . . . . . . . . . . . . . . . . . . 47
1.4.34 Declare Variable as Symetric Function (symmetric) . . . . 48
2 Numeric Evaluation 49
2.1 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1.1 Big Float (bﬀac) . . . . . . . . . . . . . . . . . . . . . . . 49
2.1.2 Big Float Zeta Function (bfhzeta) . . . . . . . . . . . . . 49
2.1.3 Convert to Big Float (bﬂoat) . . . . . . . . . . . . . . . . 49
2.1.4 Is Big Float (bﬂoatp) . . . . . . . . . . . . . . . . . . . . 50
2.1.5 Digamma Big Float Function (bfpsi) & (bfpsi0) . . . . . . 50
2.1.6 Big Float to Rational Number (bftorat) . . . . . . . . . . 50
2.1.7 Truncate Big Float (bftrunc) . . . . . . . . . . . . . . . . 50
2.1.8 Big Float Zeta Function (bfzeta) . . . . . . . . . . . . . . 50
2.1.9 Complex Big Float (cbﬀac) . . . . . . . . . . . . . . . . . 50
2.1.10 Floating Point (ﬂoat) . . . . . . . . . . . . . . . . . . . . 51
2.1.11 Float to Big Float (ﬂoat2bf) . . . . . . . . . . . . . . . . 51
2.1.12 Is Floating Number (ﬂoatnump) . . . . . . . . . . . . . . 51
2.1.13 Float Point Precision (fpprec) . . . . . . . . . . . . . . . . 51
2.1.14 Float Point Print Precission (fpprintprec) . . . . . . . . . 51
2.1.15 Keep Float Points (keepﬂoat) . . . . . . . . . . . . . . . . 51
2.1.16 Numerical Solution (numer) . . . . . . . . . . . . . . . . . 51
2.1.17 Numerical Power of Negative Integer (numer pbranch) . . 52
2.1.18 Number to Variables (numerval) . . . . . . . . . . . . . . 52
2.1.19 Tolerance in Floating Point (ratepsilon) . . . . . . . . . . 52
2.1.20 Rationalize (rationalize) . . . . . . . . . . . . . . . . . . . 52
2.1.21 Is Rational Number (ratnump) . . . . . . . . . . . . . . . 53
2.1.22 Print Message Rational (ratprint) . . . . . . . . . . . . . 53
2.1.23 Numerical Status (statsnumer) . . . . . . . . . . . . . . . 53
3 Conditions 55
3.1 Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 If Statement (if) . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 If-Else Statement (if-else) . . . . . . . . . . . . . . . . . . 56
3.2 Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 For Statement (for) . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 Do Statement (do) . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3 While Statement (while) . . . . . . . . . . . . . . . . . . . 57
4 Arithmetic 59
4.1 Arithmatic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Absolute Number (abs) . . . . . . . . . . . . . . . . . . . 59
4.1.2 Summation of Terms (sum) . . . . . . . . . . . . . . . . . 59
4.1.3 Numerical Summation of Terms (nusum) . . . . . . . . . 60
4.1.4 Numerical Factor of Algebraic Fraction (numfactor) . . . 60
4.1.5 Product of Terms (product) . . . . . . . . . . . . . . . . . 61

CONTENTS 7
4.1.6 Maximum Value (max) . . . . . . . . . . . . . . . . . . . 61
4.1.7 Minimum Value (min) . . . . . . . . . . . . . . . . . . . . 61
4.1.8 Modulo (mod) . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.9 Square Root (sqrt) . . . . . . . . . . . . . . . . . . . . . . 62
4.1.10 Sorting Elements (sort) . . . . . . . . . . . . . . . . . . . 62
4.1.11 Simpliﬁcation Environment (simp) . . . . . . . . . . . . . 62
5 Algebra 63
5.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Algebraic Evaluation (algebraic) . . . . . . . . . . . . . . 63
5.1.2 Solving Linear Equation (solve) . . . . . . . . . . . . . . . 63
5.1.3 Simplify Logs, Exponent & Radicals (radcan) . . . . . . . 64
5.1.4 Radical Expansion (radexpand) . . . . . . . . . . . . . . . 64
5.1.5 Rational Simpliﬁcation (ratsimp) . . . . . . . . . . . . . . 64
5.1.6 Rationalized Simpliﬁcation Flag (ratsimpexpons) . . . . . 65
5.1.7 Evaluation of Expression in Environment (ev) . . . . . . . 65
5.1.8 Evaluation of Expression (eval) . . . . . . . . . . . . . . . 66
5.1.9 Expansion of Algebraic Relations (expand) . . . . . . . . 66
5.2 Inequality & Polynomial . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 Equality (equal) . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Inequality (notequal) . . . . . . . . . . . . . . . . . . . . . 67
5.2.3 Solve Algebraic Polynomials (algsys) . . . . . . . . . . . . 67
5.2.4 Solve Linear Equations (linsolve) . . . . . . . . . . . . . . 68
5.2.5 Check The Stat Of Database (is) . . . . . . . . . . . . . . 68
5.2.6 Check The Stat Of Database (maybe) . . . . . . . . . . . 68
5.2.7 Algebraic Substitution (subst) . . . . . . . . . . . . . . . 69
6 Logarithm 71
6.0.8 Expoential (exp) . . . . . . . . . . . . . . . . . . . . . . . 71
6.0.9 Logarithm (log) . . . . . . . . . . . . . . . . . . . . . . . . 71
6.0.10 Absolute Logarithm (logabs) . . . . . . . . . . . . . . . . 71
6.0.11 Arc Logarithm (logarc) . . . . . . . . . . . . . . . . . . . 71
6.0.12 logconcoeﬀp . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.0.13 logcontract . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.0.14 Logarithmic Expansion (logexpand) . . . . . . . . . . . . 71
6.0.15 lognegint . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.0.16 Logarithmic Numerical Value (lognumer) . . . . . . . . . 71
6.0.17 Logarithmic Simpliﬁcation (logsimp) . . . . . . . . . . . . 71
7 Calculus 73
7.1 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.1.1 Finding Limit (limit) . . . . . . . . . . . . . . . . . . . . . 73
7.1.2 Limit Substitution (limsubst) . . . . . . . . . . . . . . . . 73
7.1.3 Limit By Taylor Series (tlimit) . . . . . . . . . . . . . . . 73
7.1.4 Taylor Limit Switch (tlimswitch) . . . . . . . . . . . . . . 74
7.2 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8 CONTENTS
7.2.1 Derivative of a Function (diﬀ) . . . . . . . . . . . . . . . . 74
7.2.2 Partial Drivative (gradef) . . . . . . . . . . . . . . . . . . 76
7.2.3 Total Diﬀerential (del) . . . . . . . . . . . . . . . . . . . . 76
7.2.4 Dependence of Function (depends) . . . . . . . . . . . . . 77
7.2.5 Show Dependencies (dependencies) . . . . . . . . . . . . . 77
7.2.6 Notation of Derivative (derivabbrev) . . . . . . . . . . . . 77
7.2.7 Value of Function at a Point (atvalue) . . . . . . . . . . . 78
7.2.8 Degree of Derivative (derivdegree) . . . . . . . . . . . . . 78
7.2.9 Partial Derivative (express) . . . . . . . . . . . . . . . . . 79
7.2.10 Selective Derivative (derivlist) . . . . . . . . . . . . . . . . 79
7.2.11 Substitution in Derivatives (derivsubst) . . . . . . . . . . 79
7.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.1 Anti Derivative of a Function (integrate) . . . . . . . . . . 79
7.3.2 Changing The Variable (changevar) . . . . . . . . . . . . 80
7.3.3 Double Integration (dblint) . . . . . . . . . . . . . . . . . 80
8 Ordinary Diﬀerential Equation 83
8.1 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . 83
8.1.1 Boundary Value Problem (bc2) . . . . . . . . . . . . . . . 83
8.1.2 Linear Ordinary Diﬀerential Equations (desolve) . . . . . 84
8.1.3 Initial Value Problem of First Order (ic1) . . . . . . . . . 84
8.1.4 Initial Value Problem of Second Order (ic2) . . . . . . . . 85
8.2 ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.2.1 Ordinary Deferential Equation of Second Order (ode2) . . 85
9 Trigonometric 87
9.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 87
9.1.1 Sine (sin) . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.1.2 Cosine (cos) . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.1.3 Tangent (tan) . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.1.4 Cotangent (cot) . . . . . . . . . . . . . . . . . . . . . . . . 88
9.1.5 Secant (sec) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.1.6 Cosecant (cosec) . . . . . . . . . . . . . . . . . . . . . . . 89
9.2 Hyperbolic Trigonometric Functions . . . . . . . . . . . . . . . . 89
9.2.1 Hyperbolic Sine (sinh) . . . . . . . . . . . . . . . . . . . . 89
9.2.2 Hyperbolic Cosine (cosh) . . . . . . . . . . . . . . . . . . 90
9.2.3 Hyperbolic Tangent (tanh) . . . . . . . . . . . . . . . . . 90
9.2.4 Hyperbolic Cotangent (coth) . . . . . . . . . . . . . . . . 90
9.2.5 Hyperbolic Secant (sech) . . . . . . . . . . . . . . . . . . 91
9.2.6 Hyperbolic Cosecant (csch) . . . . . . . . . . . . . . . . . 91
9.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . 91
9.3.1 Inverse Sine (asin) . . . . . . . . . . . . . . . . . . . . . . 91
9.3.2 Inverse Cosine (acos) . . . . . . . . . . . . . . . . . . . . . 91
9.3.3 Inverse Tangent (atan) . . . . . . . . . . . . . . . . . . . . 92
9.3.4 Inverse Tangent Type Two (atan2) . . . . . . . . . . . . . 92
9.3.5 Inverse Cotangent (acot) . . . . . . . . . . . . . . . . . . . 92

CONTENTS 9
9.3.6 Inverse Secant (asec) . . . . . . . . . . . . . . . . . . . . . 93
9.3.7 Inverse Cosecant (acsc) . . . . . . . . . . . . . . . . . . . 93
9.4 Inverse Hyperbolic Trigonometric Functions . . . . . . . . . . . . 93
9.4.1 Inverse Hyperbolic Sine (asinh) . . . . . . . . . . . . . . . 93
9.4.2 Inverse Hyperbolic Cosine (acosh) . . . . . . . . . . . . . 94
9.4.3 Inverse Hyperbolic Tangent (atanh) . . . . . . . . . . . . 94
9.4.4 Inverse Hyperbolic Cotangent (acoth) . . . . . . . . . . . 95
9.4.5 Inverse Hyperbolic Secant (asech) . . . . . . . . . . . . . 95
9.4.6 Inverse Hyperbolic Cosecant (acsch) . . . . . . . . . . . . 95
9.5 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . 96
9.5.1 Half Angle (halfangles) . . . . . . . . . . . . . . . . . . . 96
9.5.2 Expansion of Trigonometric Functions (trigexpand) . . . . 96
9.5.3 Sum of Arguments (trigexpandplus) . . . . . . . . . . . . 97
9.5.4 Product of Arguments (trigexpandtimes) . . . . . . . . . 97
9.5.5 Inverse Sequence Flag (triginverses) . . . . . . . . . . . . 98
9.5.6 Expansion of Rational Fractions (trigrat) . . . . . . . . . 98
9.5.7 Reduce Trigonometric Expressions (trigreduce) . . . . . . 99
9.5.8 Negative Arguments Simpliﬁcation (trigsign) . . . . . . . 99
9.5.9 Trigonometric Simpliﬁcation (trigsimp) . . . . . . . . . . 99
9.5.10 Demoivre Expansion (demoivre) . . . . . . . . . . . . . . 99
10 Matrix 101
10.1 Mathematical Operaton . . . . . . . . . . . . . . . . . . . . . . . 101
10.1.1 Matrix (matrix) . . . . . . . . . . . . . . . . . . . . . . . 101
10.1.2 Scalar Addition of Matrices (+) . . . . . . . . . . . . . . . 101
10.1.3 Scalar Subtraction of Matrices (-) . . . . . . . . . . . . . 102
10.1.4 Scalar Multiplication of Matrices (*) . . . . . . . . . . . . 102
10.1.5 Scalar Division of Matrix (/) . . . . . . . . . . . . . . . . 103
10.1.6 Exponent of Matrix (ˆ) . . . . . . . . . . . . . . . . . . . 103
10.1.7 Matrix Dot Product (.) . . . . . . . . . . . . . . . . . . . 104
10.2 Inverse of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.2.1 Invert of Matrix Elements (Aˆ-1) . . . . . . . . . . . . . . 105
10.2.2 Inverse of a Matrix (Aˆˆ-1) . . . . . . . . . . . . . . . . . 105
10.2.3 Append Columns (addcol) . . . . . . . . . . . . . . . . . . 105
10.2.4 Append Rows (addrow) . . . . . . . . . . . . . . . . . . . 106
10.2.5 Ajoint of Matrix (adjoint) . . . . . . . . . . . . . . . . . . 106
10.2.6 Augmented Matrix (augcoefmatrix) . . . . . . . . . . . . 106
10.2.7 Cauchy Matrix (cauchy matrix) . . . . . . . . . . . . . . . 107
10.2.8 Characteristics Polynomial (charpoly) . . . . . . . . . . . 108
10.2.9 Coeﬃcient Matrix (coefmatrix) . . . . . . . . . . . . . . . 108
10.2.10 Columns of Matrix (col) . . . . . . . . . . . . . . . . . . . 108
10.2.11 Column Vectors (columnvector) . . . . . . . . . . . . . . . 109
10.2.12 Copy Matrix (copymatrix) . . . . . . . . . . . . . . . . . . 109
10.2.13 Determinant Of Matrix (determinant) . . . . . . . . . . . 109
10.2.14 Determinant As Fraction (detout) . . . . . . . . . . . . . 109
10.2.15 Diagonal Matrix (diagmatrix) . . . . . . . . . . . . . . . . 110

10 CONTENTS
10.2.16 Do All Matrix Operations (doallmxops) . . . . . . . . . . 110
10.2.17 Do Exponent Operations (domxexpt) . . . . . . . . . . . 110
10.2.18 Do Matrix-Matrix Operations (domxmxops) . . . . . . . . 110
10.2.19 Do Matrix Products (domxnctimes) . . . . . . . . . . . . 110
10.2.20 Do Not Factorize (dontfactor) . . . . . . . . . . . . . . . . 110
10.2.21 Do Scalar Matrix Operations (doscmxops) . . . . . . . . . 111
10.2.22 Do Scalar Matrix Operation as Matrix (doscmxplus) . . . 111
10.2.23 Product of Zero & Non-Scalar Term (dot0nscsimp) . . . . 111
10.2.24 Product of Zero & Scalar Term (dot0simp) . . . . . . . . 111
10.2.25 Product of One & Other Term (dot1simp) . . . . . . . . . 111
10.2.26 Associative of Dot Product (dotassoc) . . . . . . . . . . . 111
10.2.27 Product of Constant & Other Term (dotconstrules) . . . . 111
10.2.28 Distribution in Dot Product (dotdistrib) . . . . . . . . . . 111
10.2.29 Simpliﬁed to Exponent (dotexptsimp) . . . . . . . . . . . 111
10.2.30 Zero Exponsnt (dotident) . . . . . . . . . . . . . . . . . . 112
10.2.31 Dot Screw Rules (dotscrules) . . . . . . . . . . . . . . . . 112
10.3 Matrix Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 112
10.3.1 Echelon Form of Matrix (echelon) . . . . . . . . . . . . . 112
10.3.2 Eigenvalues (eivals) . . . . . . . . . . . . . . . . . . . . . . 112
10.3.3 Eigenvectors (eivects) . . . . . . . . . . . . . . . . . . . . 113
10.3.4 Except Matrix (ematrix) . . . . . . . . . . . . . . . . . . . 113
10.3.5 User Matrix (entermatrix) . . . . . . . . . . . . . . . . . . 113
10.3.6 Generate Matrix (genmatrix) . . . . . . . . . . . . . . . . 114
10.3.7 Identity Matrix (ident) . . . . . . . . . . . . . . . . . . . . 114
10.3.8 Inverse Matrix (invert) . . . . . . . . . . . . . . . . . . . . 114
10.3.9 List Elements of Matrix (listmatrixentries) . . . . . . . 115
10.3.10 Left Side Delimiter (lmxchar) . . . . . . . . . . . . . . . . 115
10.3.11 Right Side Delimiter (rmxchar) . . . . . . . . . . . . . . . 115
10.3.12 Adjoin Matrix (adjoin) . . . . . . . . . . . . . . . . . . . . 115
10.3.13 Transpose of a Matrix (transpose) . . . . . . . . . . . . . 115
10.3.14 Mapping of Matrix (matrixmap) . . . . . . . . . . . . . . 116
10.3.15 Expression As Matrix (matrixp) . . . . . . . . . . . . . . 116
10.3.16 Trace of Matrix (mattrace) . . . . . . . . . . . . . . . . . 116
10.3.17 Minor of Matrix (minor) . . . . . . . . . . . . . . . . . . . 116
10.3.18 Determinant of Matrix (newdet) . . . . . . . . . . . . . . 117
10.3.19 Permanent of Matrix (permanent) . . . . . . . . . . . . . 117
10.3.20 Rank of Matrix (rank) . . . . . . . . . . . . . . . . . . . . 117
10.3.21 Rationalize to Matrix (ratmx) . . . . . . . . . . . . . . . . 117
10.3.22 Row of Matrix (row) . . . . . . . . . . . . . . . . . . . . . 117
10.3.23 Solve To Scalar Matrix (scalarmatrixp) . . . . . . . . . . 118
10.3.24 Sub Matrix (submatrix) . . . . . . . . . . . . . . . . . . . 118
10.3.25 Triangularization of Matrix (triangularize) . . . . . . . . . 118
10.3.26 Zero Matix (zeromatrix) . . . . . . . . . . . . . . . . . . . 118
10.3.27 Matrix Element Transpose (matrixelementtranspose) . . 118
10.3.28 Matrix Element Multiplication (matrixelementmult) . . 119
10.3.29 Matrix Element Addition (matrixelementadd) . . . . . . 119

CONTENTS 11
10.4 Solution of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11 Array 121
12 Special Functions 123
12.1 airy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.1.1 airyai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.1.2 airybi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.1.3 airydai . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.1.4 airydbi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.2 Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.2.1 Bessel Function of First Kind (besselj) . . . . . . . . . . 123
12.2.2 Bessel Function of Second Kind (bessely) . . . . . . . . . 124
12.2.3 Hyperbolic Bessel Function of First Kind (besseli) . . . . 124
12.2.4 Hyperbolic Bessel Function of Second Kind (besselk) . . 125
12.2.5 Expansion of Bessel Function (besselexpand) . . . . . . . 125
12.3 Beta Function (beta) . . . . . . . . . . . . . . . . . . . . . . . . . 125
12.4 Gamma Function (gamma) . . . . . . . . . . . . . . . . . . . . . 126
12.5 Euler Function (euler) . . . . . . . . . . . . . . . . . . . . . . . . 126
12.6 Error Function (erf) . . . . . . . . . . . . . . . . . . . . . . . . . 127
12.7 Laplace Transform (laplace) . . . . . . . . . . . . . . . . . . . . . 127
12.7.1 Inverse Laplace Transform (ilt) . . . . . . . . . . . . . . . 127
12.8 Lagrange Polynomial (lagrange) . . . . . . . . . . . . . . . . . . . 128
12.9 Laguerre Polynomial (laguerre) . . . . . . . . . . . . . . . . . . . 128
12.10Legendre Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 128
12.10.1 Legendre Polynomial of First Kind(legendrep) . . . . . . 129
12.10.2 Legendre Polynomial of Second Kind(legendreq) . . . . . 129
12.11Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.1 jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.2 jacobicd . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.3 jacobicn . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.4 jacobics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.5 jacobidc . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.6 jacobidn . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.7 jacobids . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.8 jacobinc . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.9 jacobind . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.10jacobins . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.11jacobip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.12jacobisc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.13jacobisd . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.14jacobisn . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.15inversejacobicd . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.16inversejacobicn . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.17inversejacobics . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.18inversejacobidc . . . . . . . . . . . . . . . . . . . . . . . 130

12 CONTENTS
12.11.19inversejacobidn . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.20inversejacobids . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.21inversejacobinc . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.22inversejacobind . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.23inversejacobins . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.24inversejacobisc . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.25inversejacobisd . . . . . . . . . . . . . . . . . . . . . . . 130
12.11.26inversejacobisn . . . . . . . . . . . . . . . . . . . . . . . 130
12.12fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.12.1 fourint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.12.2 fourintcos . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.12.3 fourintsin . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.12.4 foursimp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.12.5 foursin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.1 elliptice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.2 ellipticec . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.3 ellipticeu . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.4 ellipticf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.5 elliptickc . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.13.6 ellipticpi . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
13 Graphics 131
13.1 Contour Plot (contourplot) . . . . . . . . . . . . . . . . . . . . . 131
13.2 Plotting 2D Plot (plot2d) . . . . . . . . . . . . . . . . . . . . . . 132
13.3 Three Dimensional Plotting (plot3d) . . . . . . . . . . . . . . . . 133
13.4 Finding Plot Options (plotoptions) . . . . . . . . . . . . . . . . 133
13.5 Set Plot Options (setplotoption) . . . . . . . . . . . . . . . . . 135
13.6 Spherical To Cartesian (sphericaltoxyz) . . . . . . . . . . . . . 136
14 Binary Operations 137
14.1 AND Operation (and) . . . . . . . . . . . . . . . . . . . . . . . . 137
14.2 OR Operation (or) . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.3 NOT Operation (not) . . . . . . . . . . . . . . . . . . . . . . . . 137
15 File Operations 139
15.1 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
15.1.1 Append Console Transcript to File (appendﬁle) . . . . . . 139
15.1.2 Read File & Evaluate (batch) . . . . . . . . . . . . . . . . 139
15.1.3 Type of File (ﬁletype) . . . . . . . . . . . . . . . . . . . . 139
15.1.4 Evaluation From File From Directory (load) . . . . . . . . 140
15.1.5 Evaluation From File (loadﬁle) . . . . . . . . . . . . . . . 140
15.1.6 File Load Path Name (loadpathname) . . . . . . . . . . . 140
15.1.7 File Load Print Message (loadprint) . . . . . . . . . . . . 140
15.1.8 Show File Contents In Console (printﬁle) . . . . . . . . . 140
15.1.9 Evaluates Expression From File (batchload) . . . . . . . . 140

CONTENTS 13
15.2 Writing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
15.2.1 Write Transcript in File (writeﬁle) . . . . . . . . . . . . . 141
15.2.2 Write Expressions to File (stringout) . . . . . . . . . . . . 141
15.2.3 ﬁle outputappend . . . . . . . . . . . . . . . . . . . . . . 141
15.2.4 Write Output To a File (withstdout) . . . . . . . . . . . 141
15.2.5 Save A File (save) . . . . . . . . . . . . . . . . . . . . . . 142
15.2.6 Close Opened File (closeﬁle) . . . . . . . . . . . . . . . . 142
15.3 Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.3.1 ﬁlesearch . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.3.2 ﬁlesearchdemo . . . . . . . . . . . . . . . . . . . . . . . 143
15.3.3 ﬁlesearchmaxima . . . . . . . . . . . . . . . . . . . . . . 143
15.3.4 ﬁlesearchusage . . . . . . . . . . . . . . . . . . . . . . . 143
15.3.5 ﬁlesearchlisp . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.3.6 ﬁlesearchtests . . . . . . . . . . . . . . . . . . . . . . . . 144
15.3.7 ﬁletypelisp . . . . . . . . . . . . . . . . . . . . . . . . . . 144
15.4 Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
15.4.1 ﬁletypemaxima . . . . . . . . . . . . . . . . . . . . . . . 144
15.4.2 pathnamedirectory . . . . . . . . . . . . . . . . . . . . . 144
15.4.3 pathnamename . . . . . . . . . . . . . . . . . . . . . . . 144
15.4.4 pathnametype . . . . . . . . . . . . . . . . . . . . . . . . 145
16 Polynomial 147
16.1 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
16.1.1 All Roots of Polynomial (allroots) . . . . . . . . . . . . . 147
16.1.2 All Big Float Roots of Polynomial (bfallroots) . . . . . . 147
16.1.3 All Roots Within Limits (nroots) . . . . . . . . . . . . . . 148
16.1.4 Nth Root of Base (nthroot) . . . . . . . . . . . . . . . . . 148
16.1.5 Real Roots (realroots) . . . . . . . . . . . . . . . . . . . . 148
16.2 Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16.2.1 Separation of Coeﬃcients (bothcoef) . . . . . . . . . . . . 149
16.2.2 Coeﬃcient of Term (coeﬀ) . . . . . . . . . . . . . . . . . . 149
16.2.3 Divide Expression (divide) . . . . . . . . . . . . . . . . . . 149
16.2.4 Eliminate Variables (eliminate) . . . . . . . . . . . . . . . 150
16.2.5 Factor of Expression (factor) . . . . . . . . . . . . . . . . 150
16.2.6 Rearrange Factors (factorout) . . . . . . . . . . . . . . . . 151
16.2.7 Rearrange Factors of Sum (factorsum) . . . . . . . . . . . 151
16.2.8 Greatest Common Divisor (gcd) . . . . . . . . . . . . . . 151
16.2.9 Greatest Common Divisor (gcdex) . . . . . . . . . . . . . 152
16.2.10 Gaussian Factor (gfactor) . . . . . . . . . . . . . . . . . . 152
16.2.11 Highest Power (hipow) . . . . . . . . . . . . . . . . . . . . 152
16.2.12 Lowest Power (lopow) . . . . . . . . . . . . . . . . . . . . 152
16.2.13 Return The Coeﬃcient (ratcoef) . . . . . . . . . . . . . . 152
16.2.14 Remainder (remainder) . . . . . . . . . . . . . . . . . . . 153
16.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16.3.1 rat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16.3.2 Return Denominator (ratdenom) . . . . . . . . . . . . . . 153

14 CONTENTS
16.3.3 Expand Ratio (ratdenomdivide) . . . . . . . . . . . . . . 153
16.3.4 Derivative of Rational Number (ratdiﬀ) . . . . . . . . . . 153
16.3.5 Rational Expansion (ratexpand) . . . . . . . . . . . . . . 153
16.3.6 Rational Factor (ratfac) . . . . . . . . . . . . . . . . . . . 153
16.3.7 Is Rational (ratp) . . . . . . . . . . . . . . . . . . . . . . . 153
16.3.8 Rational Simpliﬁcation (ratsimp) . . . . . . . . . . . . . . 154
16.3.9 Simplify Rational Exponent (ratsimpexpons) . . . . . . . 154
16.3.10 Rational Substitution (ratsubst) . . . . . . . . . . . . . . 154
16.3.11 Resultant (resultant) . . . . . . . . . . . . . . . . . . . . . 154
17 power series 155
17.1 Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
17.1.1 Deﬁne Taylor Function (deftaylor) . . . . . . . . . . . . . 155
17.1.2 Max Taylor Order (maxtayorder) . . . . . . . . . . . . . . 155
17.1.3 Power Series (powerseries) . . . . . . . . . . . . . . . . . . 155
17.1.4 Taylor Series (taylor) . . . . . . . . . . . . . . . . . . . . . 156
17.1.5 taylor logexpand . . . . . . . . . . . . . . . . . . . . . . . 156
17.1.6 taylorordercoeﬃcients . . . . . . . . . . . . . . . . . . . 157
17.1.7 taylorsimpliﬁer . . . . . . . . . . . . . . . . . . . . . . . . 157
17.1.8 taylortruncatepolynomials . . . . . . . . . . . . . . . . . 157
17.1.9 Depth of Taylor Terms (taylordepth) . . . . . . . . . . . . 157
17.1.10 Info of Taylor Series (taylorinfo) . . . . . . . . . . . . . . 157
17.1.11 Check Series as Taylor (taylorp) . . . . . . . . . . . . . . 157
17.1.12 Print Power Series Message (verbose) . . . . . . . . . . . 157
17.1.13 Integer to Poisson (intopois) . . . . . . . . . . . . . . . . 157
17.1.14 outofpois . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
17.2 Poisson Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
17.2.1 Poisson Diﬀerentiation (poisdiﬀ) . . . . . . . . . . . . . . 158
17.2.2 Poisson Exponent (poisexpt) . . . . . . . . . . . . . . . . 158
17.2.3 Poisson Integration (poisint) . . . . . . . . . . . . . . . . 158
17.2.4 Poisson Limit (poislim) . . . . . . . . . . . . . . . . . . . 158
17.2.5 Poisson Plus (poisplus) . . . . . . . . . . . . . . . . . . . 159
17.2.6 Poisson Simpliﬁcation (poissimp) . . . . . . . . . . . . . . 159
17.2.7 Poisson Substitution (poissubst) . . . . . . . . . . . . . . 159
17.2.8 Poisson Multiplicaiton (poistimes) . . . . . . . . . . . . . 159
17.2.9 Print Poisson Series (printpois) . . . . . . . . . . . . . . . 159
18 Sets 161
18.1 Properties of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
18.1.1 Distinct Elements of The Set (cardinality) . . . . . . . . . 161
18.1.2 Disjoin of Sets (disjoin) . . . . . . . . . . . . . . . . . . . 161
18.1.3 Disjoint Sets (disjointp) . . . . . . . . . . . . . . . . . . . 162
18.1.4 Element of a Set (elementp) . . . . . . . . . . . . . . . . . 162
18.1.5 Empty Set (emptyp) . . . . . . . . . . . . . . . . . . . . . 162
18.1.6 External Subset (extremalsubset) . . . . . . . . . . . . . 162
18.1.7 ﬂatten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

CONTENTS 15
18.1.8 fulllistify . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
18.1.9 List From a Set (listify) . . . . . . . . . . . . . . . . . . . 162
18.1.10 Construct a Set (makeset) . . . . . . . . . . . . . . . . . . 163
18.1.11 Partition of Set (partitionset) . . . . . . . . . . . . . . . 163
18.1.12 Ordered Permutations of a Set (permutations) . . . . . . 163
18.1.13 Random Permutation (randompermutation) . . . . . . . 163
18.1.14 Check Set Equality (setequalp) . . . . . . . . . . . . . . . 164
18.1.15 Check For argument as Set (setp) . . . . . . . . . . . . . 164
18.2 Operations On Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 164
18.2.1 Maximum Element Value of a Set (lmax) . . . . . . . . . 164
18.2.2 Minimum Element Value of a Set (lmin) . . . . . . . . . . 164
18.2.3 Sub Set (subset) . . . . . . . . . . . . . . . . . . . . . . . 165
18.2.4 Predicate Subset (subsetp) . . . . . . . . . . . . . . . . . 165
18.2.5 Symmentric Diﬀerence (symmdiﬀerence) . . . . . . . . . . 165
18.2.6 Subtraction of Sets (setdiﬀerence) . . . . . . . . . . . . . 165
18.2.7 Cartesian Product (cartesianproduct) . . . . . . . . . . . 165
18.2.8 Intersection of Sets (intersection) . . . . . . . . . . . . . . 166
18.2.9 Power Set (powerset) . . . . . . . . . . . . . . . . . . . . . 166
18.2.10 Union Of Sets (union) . . . . . . . . . . . . . . . . . . . . 166
19 Statistics 167
19.1 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
19.1.1 Bar Plot of Data (barsplot) . . . . . . . . . . . . . . . . . 167
19.1.2 Error Box Plot of Data (boxplot) . . . . . . . . . . . . . . 168
19.1.3 Stem Plot (stemplot) . . . . . . . . . . . . . . . . . . . . . 168
19.1.4 Pie Chart (piechart) . . . . . . . . . . . . . . . . . . . . . 168
19.1.5 Scatterin Plot of Data (scatterplot) . . . . . . . . . . . . . 168
19.1.6 Asterisk Plot of Data (starplot) . . . . . . . . . . . . . . . 168
19.1.7 Histogram Plot (histogram) . . . . . . . . . . . . . . . . . 168
19.2 Regression & Correlation . . . . . . . . . . . . . . . . . . . . . . 168
19.2.1 Building a Sample (buildsample) . . . . . . . . . . . . . . 168
19.2.2 Correlation (cor) . . . . . . . . . . . . . . . . . . . . . . . 168
19.2.3 Covariance Matrix (cov) . . . . . . . . . . . . . . . . . . . 168
19.2.4 Covariance of Type One (cov1) . . . . . . . . . . . . . . . 169
19.2.5 Coeﬃcient Variation (cv) . . . . . . . . . . . . . . . . . . 169
19.2.6 List Correlation (listcorrelations) . . . . . . . . . . . . . 169
19.3 Mean, Median & Deviation . . . . . . . . . . . . . . . . . . . . . 169
19.3.1 Average or Mean (mean) . . . . . . . . . . . . . . . . . . 169
19.3.2 Mean Deviation (meandeviation) . . . . . . . . . . . . . 170
19.3.3 Median (median) . . . . . . . . . . . . . . . . . . . . . . . 170
19.3.4 Deviation from Median (mediandeviation) . . . . . . . . 171
19.3.5 Continuous Frequency (continuousfreq) . . . . . . . . . . 171
19.3.6 Discrete Frequency (discretefreq) . . . . . . . . . . . . . 171
19.3.7 Geometric Means (geometricmean) . . . . . . . . . . . . 171
19.3.8 Harmonic Mean (harmonicmean) . . . . . . . . . . . . . 172
19.3.9 Range of Data (range) . . . . . . . . . . . . . . . . . . . . 172

16 CONTENTS
19.3.10 Maximum Value of Sample List (smax) . . . . . . . . . . 172
19.3.11 Minimum Value of Sample List (smin) . . . . . . . . . . . 173
19.3.12 Standard Deviation (std) . . . . . . . . . . . . . . . . . . 173
19.3.13 Standard Deviation Type One (std1) . . . . . . . . . . . . 173
19.3.14 Transforming Samples (tranform sample) . . . . . . . . . 173
19.3.15 Variance (var) . . . . . . . . . . . . . . . . . . . . . . . . 173
19.3.16 Variance of Type One (var1) . . . . . . . . . . . . . . . . 174
19.4 Skew, Kurosis & Moment . . . . . . . . . . . . . . . . . . . . . . 174
19.4.1 Skewness of Data (skewness) . . . . . . . . . . . . . . . . 174
19.4.2 Pearson Skewness (pearsonskewness) . . . . . . . . . . . 175
19.4.3 Quartile Skewness (quartileskewness) . . . . . . . . . . . 175
19.4.4 Kurtosis (kurtosis) . . . . . . . . . . . . . . . . . . . . . . 175
19.4.5 noncentralmoment . . . . . . . . . . . . . . . . . . . . . . 176
19.4.6 Central Moment of Data (centralmoment) . . . . . . . . 176

Part I
Maxima Commands
17

Chapter 1
Maxima Core
1.1 Mathematical
1.1.1 Factorial Symbol (!)
It represents to the factorial operator. An! represents the factorial ofnin form
ofn×(n−1)×. . .×3×2×1. Factorial of negative number is inﬁnity.
✞
1(%i1) 5!
✆
The output is
✞
(%o1) 120
✆
Factorial of a fraction number in three place precision is
✞
1(%i1) 0.5!
✆
The output is
✞
(%o1) 0.886
✆
Factorial of 0.5! is
√
π/2.
1.1.2 Double Factorial Symbol (!!)
It represents to the double factorial operator. An!! represents the factorial
in form ofn×(n−2)×. . .×5×3×1. Ifnis an odd number otherwise
n×(n−2)×. . .×4×2 ifnis an even number.
✞
1(%i1) 5!!
✆
The output is
✞
(%o1) 15
✆
19

20 CHAPTER 1. MAXIMA CORE
1.1.3 Termination of Line (;)
Symbol ‘;’ (colon) is used to terminate a line in Maxima console. Every thing
written after the symbol ‘;’ is treated as comments and it is skipped by Maxima
during evaluation of transcript.
✞
1(%i1) 5!!; double factorial
✆
The output is
✞
(%o1) 15
✆
1.1.4 Negation of Expression (#)
#represents negation of syntactic equality. Note that because ofthe rules for
evaluation of predicate expressions “nota = b” is equivalent tois(a # b)instead
of “a # b”.
1.1.5 Evaluate Internally & No Output ($)
$restrict the evaluation of expression internally and the result is notprinted in
output console. This operator is always post ﬁxed to transcripts.The syntax is
✞
1(%i1) <transcript>$
✆
For example
✞
1(%i1)matrix([1,2],[3,4])$
✆
will be evaluated internally and there is no output in console. Remember that
this symbol must not followed by any string or digits or comments.
1.1.6 Do Not Execute Function (’)
If a function is preﬁxed with (’) then function is not executed. It just returns
the argument.
✞
1(%i1)’integrate(x,x)
✆
The output is
(%o1)
Z
x dx
and function without preﬁx with (’)
✞
1(%i2)integrate(x,x)
✆

1.1. MATHEMATICAL 21
The output is
(%o2)
x
2
2
1.1.7 Re-evaluation of Speciﬁc Input (“)
The double single quote (“) provides re-evaluation of speciﬁc input once again.
For example, to evaluate the ﬁrst input (%i1) once again as fourth input (%i4),
use syntax like
✞
1(%i4)’’%i1
✆
Example is
✞
1(%i2)integrate(x, x);
(%i3)integrate(x^2, x);
3(%i4)integrate(x^3, x);
(%i5)’’%i2
✆
The output is
(%o2)
x
2
2
(%o3)
x
3
3
(%o4)
x
4
4
(%o5)
x
2
2
1.1.8 Variable Assigning Operator (:)
It assign a value to a variable.
✞
(%i4) a:2
✆
It sets ‘2’ to variable ‘a’.
1.1.9 Operator Assigner (::)
::is similar to the:but it diﬀers to:by evaluation method.:evaluates only
left hand side but::evaluates left hand side as well as right hand side.

22 CHAPTER 1. MAXIMA CORE
✞
1(%i4) y:[a,b,c];%initializing operators
(%i5) y::[10,11,12];%initializing values
3(%i6) a;%call operator ’a’
✆
✞
(%o4) 10
✆
1.1.10 Macro Function Deﬁnition Operator (::=)
::=deﬁnes a function which quotes its arguments, and the expressionwhich it
returns is evaluated in the context from which the macro was called.
✞
1(%i4) g(x)::=x/99;
(%i5) g(9999);
✆
✞
(%o5) 101
✆
1.1.11 Function Operator (:=)
:=is used to initialize a function.:=never evaluates the function body. The
syntax for function operator is like
✞
1(%i4) <function name>(<independent variable>):=<function body>
✆
For example
✞
1(%i4) f(x):=sin(x)
✆
(%o4)f(x) := sin(x)
To call the function use following syntax.
✞
1(%i4) f(<independent variable value>)
✆
If user deﬁned function body consists in-build functions then result is just sym-
bolic otherwise it gives numeric result. For example
✞
1(%i4) f(x):=sin(x);
(%i5) f(10);
✆
(%o5) sin(10)
To get the numerical result rather than symbolic result then the syntax becomes
like

1.1. MATHEMATICAL 23
✞
(%i4) f(x):=sin(x);
2(%i5) ev(f(10),numer);
✆
✞
(%o5) -0.54402111088937
✆
deﬁneis also used to deﬁne a function. Its syntax is
✞
1(%i4) define(<user define function>, <function body>);
✆
1.1.12 Sum Operator (+)
It is arithmetic additive symbol. It is used to add two or more variables. If
variables are previously assigned the values then variables are replaced by their
corresponding values and resultant sum is printed in output otherwise symbolic
addition is obtained at output. It performs algebraic sum operation. It is
commutative operator and can be used in an array. Maxima sorts operands
before applying ‘+’ operator. Among ‘+’, ‘-’, ‘/’ and ‘*’, it has third highest
priority during arithmetic calculations. Scope of operator must be speciﬁed by
using parenthesis brackets.
✞
1(%i4) 10+11
✆
✞
(%o4) 21
✆
Another example is
✞
1(%i4) a:10;
(%i5) a+b
✆
(%o5)b+ 10
1.1.13 Subtraction Operator (-)
It is arithmetic subtraction symbol. It is used to subtract two or more variables.
If variables are previously assigned values then variables are replaced by their
corresponding values and resultant subtraction is printed in output otherwise
symbolic subtraction is obtained at output. It performs algebraic subtraction
operation. It is not a commutative operator. Among ‘+’, ‘-’, ‘/’ and ‘*’,it has
fourth highest priority during arithmetic calculations. Scope of operator must
be speciﬁed by using parenthesis brackets.
✞
(%i4) 10-11
✆

24 CHAPTER 1. MAXIMA CORE
(%o4) −1
Another example is
✞
1(%i4) a:10;
(%i5) a-b
✆
(%o5) −b+ 10
1.1.14 Product Operator (*)
It is arithmetic multiplicative symbol. It is used to multiply two or more vari-
ables. If variables are previously assigned with values then variablesare replaced
by their corresponding values and resultant multiplication is printed inoutput
otherwise symbolic multiplication is obtained at output. It is commutative op-
erator and can be used in an array. Maxima sorts operands beforeapplying ‘*’
operator. Among ‘+’, ‘-’, ‘/’ and ‘*’, it has second highest priority during arith-
metic calculations. Scope of operator must be speciﬁed by using parenthesis
brackets.
✞
(%i4) 10*11
✆
(%o4) 110
Another example is
✞
1(%i4) a:10;
(%i5) a*b
✆
(%o5) 10b
1.1.15 Division Operator (/)
It is arithmetic division symbol. It is used to divide two variables. If variables
are previously assigned with values then variables are replaced by their corre-
sponding values and resultant division is printed in output otherwise symbolic
division is obtained at output. It is not a commutative operator. Among ‘+’,

1.1. MATHEMATICAL 25
‘-’, ‘/’ and ‘*’, it has highest priority during arithmetic calculations. Scope of
operator must be speciﬁed by using parenthesis brackets.
✞
(%i4) 100/10
✆
(%o4) 10
Another example is
✞
1(%i4) a:10;
(%i5) a/b
✆
(%o5)
10
b
1.1.16 Exponential Operator (ˆ)
It is arithmetic exponential symbol. It is used to raise the power by anumber
over the base value. It is binary and non-commutative operator. Grouping of
terms is required to limit the scope of the operator otherwise its scope is limited
to the succeeding terms only. If base and exponent are symbols then symbolic
output is obtained and if base and exponent are assigned numericalvalue then
variables are replaced by their corresponding numerical values.
✞
(%i4) 10^2
✆
(%o4) 100
Another example is
✞
1(%i4) 10^(2^2);
✆
(%o4) 10000
✞
1(%i5) a^b;
✆
(%o5)a
b

26 CHAPTER 1. MAXIMA CORE
1.1.17 Equal Operator (=)
=denotes an equation. It also represents to the equal boolean operator. In
algebraic equations, equal sign indicates that left hand side is equalto the right
hand side for speciﬁc values of independent variable. For example
2x+ 3y= 10
If we choosex= 2 andy= 2 then left hand side of above equation is equal to
the right hand side of the equation. Symbolically, ifa=bthen valuesaandb
are equal.
1.1.18 Less Than Operator (<)
It is mathematical operator that tells whether the left hand side is lesser than
the right hand side term.
✞
1(%i4)is(a < b);
✆
✞
(%o4)unknown
✆
If terms are purely number then answer is either “true’ or ‘false’.
✞
1(%i4)is(1 < 2);
✆
✞
(%o4) true
✆
1.1.19 Grater Than Operator (>)
It is mathematical operator that tells whether the left hand side is greater than
the right hand side term.
✞
1(%i4)is(a > b);
✆
✞
(%o4)unknown
✆
If terms are purely number then answer is either “true’ or ‘false’.
✞
1(%i4)is(1 > 2);
✆
✞
(%o4) false
✆

1.1. MATHEMATICAL 27
1.1.20 Less Than or Equals To Operator (<=)
It is mathematical operator that tells whether the left hand side is lesser than
or equals to the right hand side term.
✞
1(%i4)is(a <=b);
✆
✞
(%o4)unknown
✆
If terms are purely number then answer is either “true’ or ‘false’.
✞
1(%i4)is(1 <=2);
✆
✞
(%o4) true
✆
1.1.21 Greater Than or Equals To Operator (>=)
It is mathematical operator that tells whether the left hand side is greater than
or equals to the right hand side term.
✞
1(%i4)is(a >=b);
✆
✞
(%o4)unknown
✆
If terms are purely number then answer is either “true’ or ‘false’.
✞
1(%i4)is(1 >=2);
✆
✞
(%o4) false
✆
1.1.22 Lisp Name (?)
The function name which is preﬁxed by ‘?’ signiﬁes that the name is a Lisp
name, not a Maxima name.
1.1.23 A List ([ ... ])
[ ... ]represents the beginning and end of a list or vector.
✞
1(%i4) y:[a,b,c];%initializing operators as a list
(%i5) y::[10,11,12];%initializing values as a list
3(%i6) a;%call operator ’a’ from operator list
✆
✞
(%o4) 10
✆

28 CHAPTER 1. MAXIMA CORE
1.1.24 Call Last Garbage (%)
Symbolic computation, tends to create a good deal of garbage (temporary or
intermediate results that are eventually not used), and eﬀective handling of it
can be crucial to successful completion of some programs. Symbol % is used for
calling last garbage.
✞
1(%i1) x^2+2$
(%i2) x^2-2$
3(%i3)solve(%); % calling last garbage
✆
(%o3) [x=−
√
2x=
√
2]
If % is suﬃxed by garbage handle like ‘i1’, ‘i2’ or ‘o1’, ‘o2’ etc then speciﬁc
garbage can be called.
✞
1(%i1) x^2+2$
(%i2) x^2-2$
3(%i3)solve(%i1); % calling first garbage
✆
(%o3) [x=−
√
2i, x=
√
2i]
1.1.25 Previous Statement (%%)
In compound statements, namely block, lambda, or (s1, ..., sn), %% is the value
of the previous statement.
1.1.26 Exponential (%e)
It represents to the exponential base. The series expansion of ‘%e’ is
e
x
= 1 +
x
1!
+
x
2
2!
+
x
3
3!
+. . .
Ifx= 1 then above series becomes
e= 1 +
1
1!
+
1
2!
+
1
3!
+. . .
The numerical value of ‘%e’ lies between 2<%e <3 and actually it is ap-
proximately equals to ‘2.718281828459045’. Remember whenever you need to
substitute ‘%e’ with its numerical value, set ‘%enumer’ to true by using the
following syntax
✞
1(%i1) %enumer:true;
✆

1.1. MATHEMATICAL 29
1.1.27 %etonumlog
When%etonumlogis ‘true’,rsome rational number, andxsome expression,
%e
(rlogx)
will be simpliﬁed intox
r
.
y=e
rlogx
Taking logarithm both side with natural base
log
ey= log
ee
rlogx
or
log
e
y= logx
r
Taking antilogarithm
y=x
r
It should be noted that theradcancommand also does the same transformation,
and more complicated transformations of this as well. Thelogcontractcommand
“contracts” expressions containing logarithms. By default it is ‘false’.
1.1.28 Negative Exponent as a Quotient (%edispﬂag)
When%edispﬂagis ‘true’, Maxima displays%eto a negative exponent as a
quotient. For example, %e
−x
is displayed as 1/%e
x
. By default it is ‘false’.
1.1.29 Demoivre Mode (%emode)
If%emodeis ‘true’ then Demoivre relatione
nπi
is solved by using relation
e
ix
= cos(x) +isin(x)
Remember that ‘n’ must be a whole integer. ‘i’ represents to imaginary constant.
For example
✞
1(%i1) %e^(5*%i*%pi)
✆
✞
(%o1) -1
✆
1.1.30 Exponential Numerical (%enumer)
It controls that whether ‘%e’ is to be used in output or its numericalvalue. If
%enumeris ‘false’, ‘%e’ is used other wise its numerical value 2.718281828459045
is placed.

30 CHAPTER 1. MAXIMA CORE
1.1.31 Gamma Constant (%gamma)
It is an Euler-Mascheroni constant. Its numerical value is 0.5772156649015329.
It is deﬁned as the limiting diﬀerence between the harmonic series andthe
natural logarithm
γ= lim
n→∞

n
X
k=1
1
k
−ln(n)
!
=
Z
∞
1

1
⌊x⌋
−
1
x

dx
Here,⌊x⌋represents the ﬂoor function.
1.1.32 Pi Value (%pi)
%pirepresents to the exponential value and symbolically it is equal toπ. Nu-
merical value ofπis 3.141592653589793 approximately.
✞
1(%i1)numer:true;
(%i2) %pi;
✆
✞
(%o1) 3.141592653589793
✆
1.1.33 Previous Output (%th)
%threturns the value of previous output expression identiﬁed by a number from
the current output. If there are totalnoutputs in the current session thenm
th
previous output is the (n−m)
th
output from the begining of session. Syntax
for this function is
✞
1(%i1) %th(<integer>)
✆
1.1.34 Most Recent Expression ()
is the most recent input expression.
✞
1(%i1)sin(%pi/2)$
(%i2):lisp$_;
✆
✞
(%o2) ((\%SIN) ((MQUOTIENT)$\%PI 2))
✆
1.1.35 Display String (disp)
It displays only the value of the arguments rather than equations.This is useful
for complicated arguments which don’t have names or where only thevalue of
the argument is of interest and not the name. Remember that those inputs
which are only variables are post-ﬁxed by symbol $. It instructs toMaxima to

1.2. ERRORS 31
accept the input as key-value like arguments rather than executeit and show
output. The key-value syntax may be in form of ‘key=value’ or ‘key:value’. If
dispexecuted successfully, “done” is returns as output by Maxima.
✞
1(%i1) b[1,2]:x-x^2$
(%i2) x:123$
3(%i3) disp(x, b[1,2],sin(1.0));
✆
✞
123
x-x^2
0.8414709848078965
(%o3) done
✆
1.2 Errors
1.2.1 Error (error)
The variableerroris set to a list describing the error. The ﬁrst element of error
is a format string, which merges all the strings among the arguments supplied
toerrorfunction and the remaining elements are the values of any non-string
arguments. Syntax forerrorfunction is
✞
(%i1) error(expr_1, ..., expr_n)
✆
1.2.2 Error Size (errorsize)
errorsizemodiﬁes error messages according to the size of expressions which
appear in them. If the size of an expression (as determined by the Lisp func-
tion ERROR-SIZE) is greater thanerrorsize, the expression is replaced in the
message by a symbol, and the symbol is assigned the expression. The symbols
are taken from the listerrorsyms. The default value oferrorsizeis ‘10’.
1.2.3 Error Symbol (errorsyms)
In error messages, expressions larger thanerrorsizeare replaced by symbols,
and the symbols are set to the expressions. The symbols are takenfrom the
listerrorsyms. The ﬁrst too-large expression is replaced byerrorsyms[1], the
second byerrorsyms[2], and so on.
1.2.4 Error Message (errormsg)
Reprints the most recent error message. The variable error holdsthe message,
anderrormsgformats and prints it.

32 CHAPTER 1. MAXIMA CORE
1.2.5 Debug Mode Maxima (debugmode)
By default, Maxima does not debugs to any errors and mistakes. Toset the
debug mode true usedebugmodefunction like
✞
1(%i4) debugmode(true)
✆
It returns ‘true’ if debug mode is set to true otherwise returns false.
1.2.6 Show Examples (example)
It returns the method for using a function whose name is supplied asan argu-
ment. To see all the listed functions for functionexampleuse it like
✞
1(%i1) example();
✆
To see the example ofnrootsuse it like
✞
1(%i1) example("nroots");
✆
It returns the method of use of functionnroots. At the end of successful execu-
tion ofexamplefunction, “done” is printed in output console.
1.2.7 Set Numerical Precision (fpprec)
fpprecis used to set the decimal places of decimal numbers ie precision of nu-
merical output. The syntax is
✞
1(%i1) fpprec : <integer for decimal places>;
✆
For example
✞
1(%i1) fpprec : 16
✆
Now the outputs will have sixteen decimal places. If we want to get result upto
ﬁve decimal places then set precision to ﬁve like
✞
1(%i1) fpprec : 5
✆
1.2.8 Toggle to Floating Point (numer)
By defalutnumeris ‘false’ and the symbols are used in calculations of expression.
Ifnumeris ‘true’ then variables or symbols in expression which have numerical
values are replaced by their values. The syntax is used like
✞
1(%i1) numer:true % for setting true value
(%i2) numer:false % for setting false value
✆
For example

1.3. PROCEDURAL 33
✞
(%i1) numer:true;
2(%i2) %e
✆
✞
(\%o2) 2.718281828459045
✆
Ifnumeris set to ‘false’ then result is symbolic.
✞
1(%i1) numer:false;
(%i2) %e
✆
✞
(%o2) %e
✆
1.3 Procedural
1.3.1 Interrupt Maxima (kill)
It removes all bindings (value, function, array, or rule) from the arguments. An
argument may be a symbol or a single array element. The syntax is
✞
1(%i1) kill(<symbol>)
(%i2) kill([<array element 1>, <array element 2>, ...)
3(%i3) kill(<labels>)
(%i4) kill(<integers>)
5(%i5) kill(all)%kills all values, function, array or rules
✆
1.3.2 Modes of Variables (modedeclare)
modedeclareis used to declare the modes of variables and functions for subse-
quent translation or compilation of functions.modedeclareis typically placed
at the beginning of deﬁnition of a function, at the beginning of a Maxima script,
or executed at the interactive prompt. The syntax used is
✞
1(%i1) mode_declare ([y_1, y_2, ...], fixnum)%Or
✆
or
✞
1(%i2) mode_declare (y_1, mode_1, ... y_n, mode_n])
✆
The ‘ﬁxnum’ are “ﬂoat”, “bignum” etc. For example,
✞
1(%i1) F(x):=(mode_declare([x],float), x+2.0)
✆
This converts independent variablexas ﬂoating value and as well as function
F(x) =x+ 2.0 into ﬂoating function.

34 CHAPTER 1. MAXIMA CORE
1.3.3 Evaluate & Print Expression (print)
printevaluates and displays expressions one after another, from left to right,
starting at the left edge of the console display. The syntax for thefunctionprint
is
✞
1(%i1)print(<expr 1>, <expr 2>, ...., <expr n>)
✆
✞
1(%i1)print((a+b)^2, (a-b)^2)
✆
(%o3)b
2
−2a b+a
2
1.3.4 Labels (labels)
labelsreturns the list of either input or output or intermediate expression labels
which begin with symbol. Typically symbol is the value of eitherincharor
outcharorlinechar. Syntax for the function is
✞
1(%i1)labels(<symbol>)
✆
1.3.5 Line Number (linenum)
It is the line number of the current pair of input and output expressions.
1.3.6 List of Options (myoptions)
myoptionsis the list of all options ever reset by the user, whether or not they
get reset to their default value.
1.3.7 Ordinary Base (obase)
obasemay be assigned any integer between 2 and 36 (decimal), inclusive. The
default value ofobasevariable is decimal ‘10’. Other bases are binary ‘2’, octal
‘8’, hexadecimal ‘16’ and ‘32’.
✞
1(%i1)obase:2$
(%i2) 99;
✆
✞
(%o2) 1100011
✆

1.3. PROCEDURAL 35
1.3.8 Print the Reset Option (optionset)
Whenoptionsetis ‘true’, Maxima print outs a message whenever a Maxima
option is reset. This is useful if the user is doubtful of the spelling ofsome
option and wants to make sure that the variable he assigned a value to was
truly an option variable. The default value is ‘false’.
1.3.9 Play Previous Garbage (playback)
It prints the previously collected garbage by the Maxima. The syntax is
✞
1(%i1)playback([<inputs>, <outputs>],grind,time, slow)
✆
Argument “slow” is useful in conjunction with save orstringoutwhen creating a
secondary-storage ﬁle in order to pick out useful expressions. Argument “time”
displays the computation time for each expression. Argument “grind” displays
input expressions in the same format as the “grind”. Output expressions are
not aﬀected by the “grind” option.
1.3.10 Prompting Symbol (prompt)
promptis the prompt symbol of thedemofunction, playback (slow) mode and
the Maxima break loop. The default value is ‘’.
1.3.11 Quit the Session (quit)
quitterminates the current Maxima session. It is used likequit().
1.3.12 Rationalized Print (ratprint)
Whenratprintis true, a message informing the user about the conversion of
ﬂoating point numbers into rational numbers is displayed. The default value is
‘true’.
1.3.13 Enter into Lisp System (tolisp)
tolispvariable let to enter into Lisp system of Maxima. It enables to initiated
new variable, function etc in Maxima that can be used later in Maxima after
returning to Maxima. To return to Maxima,
✞
1(%i1) to_maxima;
✆
variable is used (including parenthesis brackets).

36 CHAPTER 1. MAXIMA CORE
1.3.14 Call Functions of Current Session (backtrace)
backtracefunction is used to call the all functions of current session. The length
of tracing is controled by supplying an integer value as argument to the function.
The syntax used is
✞
1(%i1) backtrace(<lengthoftraceas integer>)
✆
1.3.15 Break Maxima Evaluation (break)
This function has syntax like
✞
1(%i1) break(<expr 1>, <expr 2>, ..., <expr n>)
✆
breakevaluates and prints ‘expr1’ to ‘exprn’ and then causes a Maxima break
at which point the user can examine and change his environment. Upon typing
“exit”; the computation resumes.
1.3.16 Bug Reporting Info (bugreport)
It returns the details of Maxima and the link where a bug can be reported.
1.3.17 Build Information (buildinfo)
buildinforeturns the information consisting date, time, building plateform,
Lisp compiler, version of Maxima and host etc used during the compilation and
building of Maxima.
1.3.18 Debug Mode (debugmode)
By defaultdebugmodeis false. When a Maxima error occurs, Maxima will start
the debugger ifdebugmodeis ‘true’. The user may enter commands to examine
the call stack, set breakpoints, step through Maxima code, and so on. Enabling
debugmode will not catch Lisp errors.
1.3.19 Show Evaluation Time (showtime)
showtimeis ‘true’ then time taken by maxima to evaluate the transcript is shown
at the output. By default it is ‘false’. This variable is useful to check the speed
of execusion of function and provide the ways of script cleaning.
1.3.20 Evaluation Time (time)
timereturns the time taken durign the evaluation by Maxima. The handle is
supplied as argument to get the time taken by Maxima in evaluation of speciﬁc
function.timecan only be applied to output line variables; for any other vari-
ables, time returns unknown. Setshowtimeas ‘true’ to make Maxima print out
the computation time and elapsed time with each output line.

1.3. PROCEDURAL 37
✞
1(%i1)integrate(x,x)
(%i2)time(%o1);
✆
✞
(%o2) [0.0]
✆
1.3.21 Promp & Read (read)
readﬁrst prints the statement and then ask for one input in the console. The
expression entered in the console is read by Maxima and returns theevaluated
result of the expression in the console. The expression entered into the console
muste be terminated by dollar sign(‘$’).
✞
1(%i1) a=10;
(%i2) f:read("Enter new value")$
3<Enter new value> 2^a$
(%i3) f;
✆
✞
(%o3) 1024
✆
1.3.22 testsuiteﬁles
Similar to theruntestsuiteto check the test pass of a ﬁle.
✞
1(%i1) testsuite_files();
✆
1.3.23 Test The Maxim (run testsuite)
runtestsuitetests producing the desired answer are considered “passes” as are
tests that do not produce the desired answer, but are marked asknown bugs.
✞
1(%i1) run_testsuite();
✆
1.3.24 Read Function Without Evaluation (readonly)
Ifreadonlyfunction is used in place ofreadfunction with similar syntax, then it
returns the expression to console without evaluation.
✞
1(%i1) a=10;
(%i2) f:readonly("Enter new value")$
3<Enter new value> 2^a$
(%i3) f;
✆
(

38 CHAPTER 1. MAXIMA CORE
1.3.25 Activate Contexts (activate)
It activates the contexts. The facts in these contexts are thenavailable to make
deductions and retrieve information. The facts in these contextsare not listed
byfacts. Syntax foractivateis
✞
(%i1) activate(<context 1>, <context 2>, ...)
✆
1.3.26 List of Activated Contexts (activecontexts)
activecontextsis a list of the contexts which are active by way of theactivate
they are subcontexts of the current context.
1.3.27 Additive Function (additive)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as a additive function. The syntax for this is
✞
1(%i1) declare(<f>, additive)
✆
1.3.28 Alternative Name for Function (alias)
It assigns an alternative name to Maxima function. Syntax for this function is
✞
1(%i1) alias(<new fcn name 1>, <old fcn name 1>, ...)
✆
1.3.29 Declare String as Alphabic (alphabetic)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
all characters in the ‘s’ as alphabatic characters. The syntax forthis ﬂag is
✞
1(%i1) declare(<s>, alphabatic)
✆
1.3.30 Arguments In Expression (args)
It returns the number of arguments in an expression. Syntax used is
✞
1(%i1) args(<expression>)
✆
1.3.31 Atomic Expression (atom)
If the function is used like
✞
1(%i1) atom(<expression>)
✆

1.3. PROCEDURAL 39
and Maxima returns ‘true’ then ‘expression’ is atomic (i.e. a number,name or
string) otherwise ‘expression’ is non atomic. Thus
✞
1(%i1) atom(5)
✆
is true while
✞
1(%i1) atom(a[1])
✆
and
✞
1(%i2) atom(sin(x))
✆
are ‘false’ (assuminga[1] andxare unbound).
1.3.32 Declare a Function Antisymmetric (antisymmet-
ric)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as an antisymmetric function. The syntax for this ﬂag is
✞
1(%i1) declare(<f>, antisymmetric)
✆
Here function ‘f’ will assumed as an antisymmetric function.
1.3.33 Check The Integer Type (askinteger)
✞
1(%i1) askinteger(<expression>, <integer type>)
✆
attempts to determine from the assume database whether expression is an in-
teger or not. There are three categories of ‘integer type’ - (i) “odd”, (ii) “even”
and (iii) “integer”.
1.3.34 Retrive User Property (get)
✞
1(%i1)get(<a>, )
✆
It retrieves the user property indicated byiassociated with atomaor returns
‘false’ if a doesn’t have propertyi.
1.3.35 Declare Variable as Integer (integer)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as an integer. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, integer)
✆
Here variable ‘var’ will assumed as an integer.

40 CHAPTER 1. MAXIMA CORE
1.3.36 Declare Variable as Non-Integer (noninteger)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as a noninteger. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, noninteger)
✆
Here variable ‘var’ will assumed as a noninteger.
1.3.37 Integer Variabe (integervalued)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as an integervalued function. The syntax for this ﬂag is
✞
1(%i1) declare(<f>, integervalued)
✆
Here function ‘f’ will assumed as an integervalued function.
1.3.38 Query for Sign of Expression (asksign)
asksignattempts to deduce the sign of expressions from the sign of operands
within the expression. For example, ifaandbare both positive, thena+bis
also positive.
✞
1(%i1) asksign(1-2)
✆
✞
(%o1) neg
✆
1.3.39 Declare Variables (assume)
It adds predicates to the current context. If a predicate is inconsistent or redun-
dant with the predicates in the current context, it is not added to the context.
The context accumulates predicates from each call to assume. Syntax for the
functionassumeis
✞
1(%i1) assume(<var> <condition> <varvalues>, .....)
✆
‘conditions’ are the mathematical comparision symbols like ‘<=’, ‘>=’, ‘<’, ‘>’
and ‘=’ etc. For example
✞
1(%i1) assume(x>1,y<1)$
(%i2)is(x>y)
✆
✞
(%o2) true
✆

1.3. PROCEDURAL 41
1.3.40 Declare Variable as Positive (assumepos)
Whenassumeposis ‘true’ and the sign of a parameterxcannot be determined
from the current context or other considerations,signandasksign (x)return
‘true’. By default it is ‘false’.
1.3.41 Declare As Communative Function (commutative)
Functiondeclare, if used as the syntax given below, tells the simpliﬁer that
function is a commutative function. The syntax for this is
✞
1(%i1) declare(<f>, commutative)
✆
1.3.42 Compare Expression (compare)
It returns the comparative result according to the comparative relation between
two supplied arguments to the functioncompare. Syntax for this function is
✞
1(%i1) compare(<expr 1>, <expr 2>)
✆
If Maxima is unable to compare then it returns “unknown”.
1.3.43 Context (context)
contextnames the collection of facts maintained byassume,forget. Binding con-
text to a name ‘foo’ changes the current context to ‘foo’. If the speciﬁed context
‘foo’ does not yet exist, it is created automatically by a call tonewcontext. The
speciﬁed context is activated automatically. By default it is “initial”.
1.3.44 Contexts (contexts)
contextsis a list of the contexts which currently exist, including the currently
active context. By default its value is [“initial”, “global”].
1.3.45 Deactivate Context (deactivate)
deactivatedeactivates the contexts supplied to the function as its arguments.
✞
1(%i1) deactivate(<context 1>, <context 2>, .....)
✆
1.3.46 Delcaration Statement (declare)
It assigns the atom or list of atoms the property or list of properties respectively.
It is used like
✞
1(%i1) declare(<atom>, <property>)
✆

42 CHAPTER 1. MAXIMA CORE
1.4 Declarations
1.4.1 Declare As Decreasing Function (decreasing)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as a decreasing function. The syntax for this is
✞
1(%i1) declare(<f>, decreasing)
✆
1.4.2 Declare As Increasing Function (increasing)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as a increasing function. The syntax for this is
✞
1(%i1) declare(<f>, increasing)
✆
1.4.3 Declare Variable as Even Variable (even)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a symbol as a even integer variable. The syntax for this is
✞
1(%i1) declare(<sym>, even)
✆
1.4.4 Declare Variable as Odd Variable (odd)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a symbol as a odd integer variable. The syntax for this is
✞
1(%i1) declare(<sym>, odd)
✆
1.4.5 Variable As a Constant (constant)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a symbol as a constant. The syntax for this is
✞
1(%i1) declare(<sym>, constant)
✆
1.4.6 Check a Constant (constantp)
It returns ‘true’ if an expression is a constant expression, otherwise returns
‘false’.
✞
1(%i1) constantp(<expression>)
✆

1.4. DECLARATIONS 43
1.4.7 Get Features (featurep)
It attempts to determine whether the object has the feature onthe basis of the
facts in the current database. If so, it returns ‘true’, else ‘false’.
✞
1(%i1)featurep(<object name>, <featurename>)
✆
1.4.8 Set Feature (features)
Maxima recognizes certain mathematical properties of functions and variables.
These are called ”features”. The syntax
✞
1(%i1) declare(x, foo)
✆
gives the propertyfooto the function or variablex. Thedeclarefunction used
like
✞
1(%i1) declare(<featurename>,feature)
✆
declares a new feature.featuresis a list of mathematical features.
1.4.9 Status (is)
isasks whether the predicate expression is provable from the factsin the as-
sume database. If predicted expression is provable, it returns ‘true’ otherwise it
returns ‘false’.
✞
1(%i1) 1<2$% itistrue as 1isalways less than 2.
(%i2)is(1<2) % ask the prediction, resultistrue
✆
✞
(%o2) true
✆
1.4.10 Kill Contextx (killcontext)
It kills the contenxt from the Maxima. Syntax is
✞
1(%i1)killcontext(<context 1>, ...)
✆
1.4.11 Simplify as Left-Associative (lassociative)
Functiondeclare, if used as the syntax given below, tells to simpliﬁer that the
function is a left-associative. The syntax for this is
✞
1(%i1) declare(<expr>,lassociative)
✆

44 CHAPTER 1. MAXIMA CORE
1.4.12 Recognize Function as N-Array (nary)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as n-array function. The syntax for this is
✞
1(%i1) declare(<f>, nary)
✆
1.4.13 Maximum Negative Expansion (maxnegex)
maxnegexis the largest negative exponent which will be expanded by the expand
command. By default its value is ‘1000’.
1.4.14 Match Declaration (matchdeclare)
Associates a predicate with a variable so that variable matches expressions for
which the predicate returns anything other than false. Syntax is
✞
1(%i1)matchdeclare(<var>, <predicate>)
✆
1.4.15 Simplify as Multiplicative (multiplicative)
Functiondeclare, if used as the syntax given below, tells Maxima to expand an
expression as a multiplicative. The syntax for this is
✞
1(%i1) declare(<expr>,multiplicative)
✆
1.4.16 Multiply Two Expression (multthru)
This function multiply two or more expressions with each other.multthrudoes
not expand exponentiated sums if present in the expressions. Thisfunction is
the fastest way to distribute products (commutative or noncommutative) over
sums.
✞
1(%i1)multthru(<expr 1>, <expr 2>)
✆
1.4.17 Negative Distribution (negdistrib)
If this variable is set to ‘true’ this negative coeﬃcient of an expression is dis-
tributed to the all terms of the expression. By default it is ‘true’.
✞
1(%i1) -(a+b)
✆
(%o1) −b−a
Now, settingnegdistribas ‘false’.

1.4. DECLARATIONS 45
✞
1(%i1)negdistrib:false$
(%i2) -(a+b)
✆
(%o2) −(b+a)
1.4.18 Create New Context (newcontext)
It creates a new contenxt in the Maxima. Syntax is
✞
(%i1)newcontext(<context>)
✆
1.4.19 Numerical Value of Variable (numerval)
It declares that a variable have numerical value.
✞
1(%i1)numerval(<var 1>, <num1>, ....., <var n>, <numn>)
✆
1.4.20 Pull Out Constant Term (outative)
Functiondeclare, if used as the syntax given below, tells Maxima to pull out
the constant term from the expression as coeﬃcient. The syntaxfor this is
✞
1(%i1) declare(<f>,outative)
✆
1.4.21 Positive Function (posfun)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a functionfto be a positive function.
✞
1(%i1)is(f(x) > 0)
✆
yields ‘true’.
✞
1(%i1) declare(<f>,posfun)
✆
1.4.22 Properties of Atom (properties)
Returns a list of the names of all the properties associated with theatom. It is
used like
✞
1(%i1)properties(<atom name>)
✆

46 CHAPTER 1. MAXIMA CORE
1.4.23 Assign a Value to Property (put)
putassigns value to the property (speciﬁed by indicator) of atom. ‘indicator’
may be the name of any property, not just a system-deﬁned property. The
syntax used is
✞
1(%i1)put(<atom>, <value>, <indicator>)
✆
1.4.24 Simplify as Right-Associative (rassociative)
Functiondeclare, if used as the syntax given below, tells to simpliﬁer that the
function is a a right-associative. The syntax for this is
✞
1(%i1) declare(<expr>,rassociative)
✆
1.4.25 Declare Variable as Irrational Variable (irrational)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as an irrational variable. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, irrational)
✆
Here variable ‘var’ will assumed as an irrational variable.
1.4.26 Declare Variable as Irrational Variable (rational)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as a rational variable. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, rational)
✆
Here variable ‘var’ will assumed as a rational variable.
1.4.27 Declare Variable as Complex Variable (complex)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as a complex variable. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, complex)
✆
Here variable ‘var’ will assumed as a complex variable.
1.4.28 Declare Variable as Imaginary Variable (imaginary)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as an imaginary variable. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, imaginary)
✆
Here variable ‘var’ will assumed as an imaginary variable.

1.4. DECLARATIONS 47
1.4.29 Declare Variable as Real Variable (real)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as a real variable The syntax for this ﬂag is
✞
1(%i1) declare(<var>, real)
✆
Here variable ‘var’ will assumed as a real variable.
1.4.30 Remove Indicator From Atom (rem)
The syntax for theremis
✞
1(%i1)rem(<atom>, <indicator>)
✆
It removes the property indicated by ‘indicator’ from ‘atom’.remreverses the
eﬀect ofput.remreturns done if atom had an indicator property whenremwas
called, or ‘false’ if it had no such property.
1.4.31 Remove Property (remove)
It removes the properties from the atom. It has syntax like
✞
1(%i1)rem(<atom vector>, <property vector>)
✆
If “all’ is used as argument in place of ‘atom vector’ then ‘property vector’ will
be removed from all atoms.
1.4.32 Scalar Value (scalar)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a variable as a scalar variable. The syntax for this ﬂag is
✞
1(%i1) declare(<var>, scalar)
✆
Here variable ‘var’ will assumed as a scalar variable.
1.4.33 Sign of Expression (sign)
signattempts to deduce the sign of expressions from the sign of operands within
the expression. For example, ifaandbare both positive, thena+bis also
positive.
✞
1(%i1)sign(1-2)
✆
✞
(%o1) neg
✆

48 CHAPTER 1. MAXIMA CORE
1.4.34 Declare Variable as Symetric Function (symmetric)
Functiondeclare, if used as the syntax given below, tells Maxima to recognize
a function as a symmetric function. The syntax for this ﬂag is
✞
1(%i1) declare(<f>,symmetric)
✆
Here function ‘f’ will assumed as a symmetric function.

Chapter 2
Numeric Evaluation
2.1 Number Theory
2.1.1 Big Float (bﬀac)
bﬀacis acronym of “Bigﬂoat” version of the factorial function. The syntax for
the function is
✞
1(%i1) bffac(<expression>, <integer>)
✆
The second argument ‘integer’ is how many digits to retain and return, it’s a
good idea to request a couple of extra.
2.1.2 Big Float Zeta Function (bfhzeta)
bfhzetareturns the Hurwitz zeta function for the argumentssandh. The return
value is a big ﬂoat (bﬂoat);nis the number of digits in the return value. The
Hurwitz zeta function is deﬁned as
zeta(s, h) =
∞
X
n=0
1
(n+h)
s
To use this function, “bﬀac” should be loaded at ﬁrst. The syntax for this
function is
✞
1(%i1) bfhzeta(<expr s>, <expr h>, <numofreturndigit n>)
✆
2.1.3 Convert to Big Float (bﬂoat)
It converts all numbers and functions of numbers in expression tobigﬂoatnum-
bers. The number of signiﬁcant digits in the resultingbigﬂoatsis speciﬁed by
the global variablefpprec.
✞
1(%i1) bfloat(<expression>)
✆
49

50 CHAPTER 2. NUMERIC EVALUATION
2.1.4 Is Big Float (bﬂoatp)
Returns ‘true’ if expression is abigﬂoatnumber, otherwise ‘false’.
✞
1(%i1) bfloatp(<expression>)
✆
2.1.5 Digamma Big Float Function (bfpsi) & (bfpsi0)
bfpsiis the polygamma function of real argumentzand integer ordern.bfpsi0
is the digamma function. The syntax is
✞
1(%i1) bfpsi0 (z,fpprec)
✆
or
✞
1(%i1) bfpsi (0, z,fpprec)
✆
These functions are same and return bigﬂoat values.fpprecis the bigﬂoat
precision of the return value.
2.1.6 Big Float to Rational Number (bftorat)
bftoratcontrols the conversion ofbﬂoatsto rational numbers. Whenbftorat
is ‘false’,ratepsilonconversion take place. Whenbftoratis ‘true’, generated
rational number generated will accurately represent thebﬂoat. By default it is
‘false’.
2.1.7 Truncate Big Float (bftrunc)
bftrunccauses trailing zeroes in non-zerobigﬂoatnumbers not to be displayed.
Thus, ifbftruncis ‘false’,bﬂoat(1)displays as 1.000000000000000B0. Otherwise,
this is displayed as 1.0B0. By default it is ‘true’.
2.1.8 Big Float Zeta Function (bfzeta)
It returns the Riemann zeta function for the arguments. The return value is a
big ﬂoat (bﬂoat);nis the number of digits in the return value. The syntax is
✞
1(%i1) bfzeta(s, n)
✆
2.1.9 Complex Big Float (cbﬀac)
It returns complex big ﬂoat and needs to loadbﬀac. The syntax is
✞
1(%i1) cbffac(<complex number>, <precision points>)
✆

2.1. NUMBER THEORY 51
2.1.10 Floating Point (ﬂoat)
It converts integers, rational numbers and bigﬂoats in expression to ﬂoating
point numbers.
2.1.11 Float to Big Float (ﬂoat2bf)
Whenﬂoat2bfis ‘false’, a warning message is printed when a ﬂoating point
number is converted into a bigﬂoat number. By default it is ‘true’.
2.1.12 Is Floating Number (ﬂoatnump)
It returns ‘true’ if expression is a ﬂoating point number, otherwise‘false’. The
syntax is
✞
1(%i1)floatnump(<expression>)
✆
2.1.13 Float Point Precision (fpprec)
fpprecis the number of signiﬁcant digits for arithmetic on bigﬂoat numbers.
fpprecdoes not aﬀect computations on ordinary ﬂoating point numbers. By
default its value is ‘16’. It is used like
✞
1(%i1)fpprec:<integer less than 16>
✆
2.1.14 Float Point Print Precission (fpprintprec)
fpprintprecis the number of digits to print when printing an ordinary ﬂoat or
bigﬂoat number. For ordinary ﬂoating point numbers, whenfpprintprechas a
value between ‘2’ and ‘16’ (inclusive), the number of digits printed is equal to
fpprintprec. Otherwise,fpprintprecis ‘0’, or greater than ‘16’, and the number
of digits printed is ‘16’. By default its value is ‘0’.
✞
1(%i1)fpprintprec:<integer 2 to 16 & 0>
✆
2.1.15 Keep Float Points (keepﬂoat)
By default it is ‘false’. Whenkeepﬂoatis ‘true’, prevents ﬂoating point numbers
from being rationalized when expressions which contain them are converted to
canonical rational expression (CRE) form.
2.1.16 Numerical Solution (numer)
numercauses some mathematical functions (including exponentiation) with nu-
merical arguments to be evaluated in ﬂoating point. It causes variables in ex-
pression which have been given numerals to be replaced by their values. It also
sets theﬂoat.

52 CHAPTER 2. NUMERIC EVALUATION
2.1.17 Numerical Power of Negative Integer (numerpbranch)
By default, this variable is ‘false’. The option variablenumerpbranchcontrols
the numerical evaluation of the power of a negative integer, rational, or ﬂoating
point number. Whennumerpbranchis ‘true’ and the exponent is a ﬂoating
point number or the option variablenumerthe numerical result returns us-
ing the principal branch, otherwise a simpliﬁed, but not an evaluatedresult is
returned.
✞
1(%i1) (-2)^0.75;
✆
✞
(%o1) (-2)^0.75
✆
✞
1(%i2) (-2)^0.75,numer_pbranch:true;
✆
✞
(%o2) 1.189207115002721*%i-1.189207115002721
✆
2.1.18 Number to Variables (numerval)
This function assigns the numerical values to the variables. Syntaxused like
✞
1(%i1)numerval(var_1, val_1, ..., var_n, val_n)
✆
2.1.19 Tolerance in Floating Point (ratepsilon)
ratepsilonis the tolerance used in the conversion of ﬂoating point numbers
to rational numbers, when the option variablebftorat. By default its value is
2.0e
−15
.
2.1.20 Rationalize (rationalize)
Convert all double ﬂoats and big ﬂoats in the Maxima expression to their exact
rational equivalents. If you are not familiar with the binary representation of
ﬂoating point numbers, you might be surprised that
✞
1(%i1) rationalize (0.1)
✆
does not equal 1/10. This behavior isn’t special to Maxima. The number 1/10
is a repeating, not a terminating, binary representation.
✞
1(%i1) rationalize(0.1)
✆
The output is
(%o1)
3602879701896397
36028797018963968

2.1. NUMBER THEORY 53
2.1.21 Is Rational Number (ratnump)
Returns ‘true’ if expression is a literal integer or ratio of literal integers, other-
wise ‘false’.
2.1.22 Print Message Rational (ratprint)
Whenratprintis ‘true’, a message informing the user of the conversion of ﬂoating
point numbers to rational numbers is displayed. By default it is ‘true’.
2.1.23 Numerical Status (statsnumer)
Ifstatsnumeris ‘true’, inference statistical functions return their results in
ﬂoating point numbers. If it is ‘false’, results are given in symbolic and rational
format.

54 CHAPTER 2. NUMERIC EVALUATION

Chapter 3
Conditions
Following conditions can be used in Maxima.
3.1 Linear
3.1.1 If Statement (if)
ifis a conditional statement. The syntax for functionifis like
✞
1(%i2)if< condition >
then
3 <bodyifcondition true>
else
5 <bodyifcondition false>
✆
✞
1(%i2) f(x):= ifx=1 then (x+1)$
(%i2) f(3)
✆
(%o1)false
The output is false asx6= 1. Again
✞
(%i2) f(x): = ifx=1 then (x+1)$
2(%i2) f(1)
✆
(%o1) 2
55

56 CHAPTER 3. CONDITIONS
3.1.2 If-Else Statement (if-else)
if-elseis a conditional statement where function body for ‘true’ and ‘false’ con-
ditions are given and only ‘true’ case is printed in output. Syntax forif-else
condition is
✞
(%i2)if< condition >
2 then
<bodyifcondition true>
4 else
<bodyifcondition false>
✆
✞
1(%i2) f(x): = ifx=1 then (x+1) else (x-1)$
(%i2) f(3)
✆
(%o1) 2
3.2 Loop
3.2.1 For Statement (for)
It is loop operator. The syntax forforloop is
✞
(%i1)for<initial var>
2 thru
<boundinglimit>
4 step
<integer>
6 do
<function body>;
✆
For example
✞
1(%i1)fori:1 thru 10 step 3 doprint(2*i);
✆
✞
2
8
14
20
(%o1) done
✆
whilestatement can be applied withforfunction.
✞
1(%i1)fori:1 while i <=10 do s:s+i;
✆
✞
(%o1) 55
✆

3.2. LOOP 57
3.2.2 Do Statement (do)
Thedostatement is used for performing iteration.
✞
1(%i1)fori:1 while i <=10 do s:s+i;
✆
✞
(%o1) 55
✆
3.2.3 While Statement (while)
It allow the iteration to be performed until the require condition is satisﬁed.
Generallywhilestatement comes withdostatement.
✞
1(%i1)fori:1 while i <=10 do s:s+i;
✆
✞
(%o1) 55
✆

58 CHAPTER 3. CONDITIONS

Chapter 4
Arithmetic
Arithmetical functions are followings.
4.1 Arithmatic Functions
4.1.1 Absolute Number (abs)
absreturns the absolute value of an supplied number. Ifx= 1.20152 then its
absolute value is 1.201. Similarly ifx=−1.20152, then its absolute value is
1.201.
✞
1(%i1) abs(1.20152);
✆
✞
(\%o1) 1.201
✆
✞
1(%i1) abs(-1.20152);
✆
✞
(\%o1) 1.201
✆
4.1.2 Summation of Terms (sum)
sumis used for sum evaluation of a series. Syntax forsumis
✞
1(%i1) sum(<expression>, <variable>, <initial value>, <final value>)
✆
Summation
n
X
i=1
i
2
is expanded as
n
X
i=1
i
2
= 1
2
+ 2
2
+ 3
2
+. . .+n
2
59

60 CHAPTER 4. ARITHMETIC
If either initial or ﬁnal value is symbol, then output is just symbolic representa-
tion of summation. The sum of a series of square of ﬁrst ‘n’ integersis evaluated
as
✞
1(%i1) sum(x^2, x, a, b)
✆
The output is
(%o1)
b
X
x=a
x
2
If initial & ﬁnal values are integers then
✞
1(%i1) sum(x^2,x,1,10)
✆
The output is
✞
(%o1) 385
✆
4.1.3 Numerical Summation of Terms (nusum)
nusumis used for numerical sum evaluation of a series. Syntax fornusumis
✞
1(%i1) nusum(<expression>, <variable>, <initial value>, <final value>)
✆
The numerical sum of a series of square of ﬁrst ‘n’ integers is evaluated as
✞
1(%i1) nusum(x^2,x,1,n)
✆
The output is
(%o1)
n(n+ 1) (2n+ 1)
6
This output is in symbolic form.
4.1.4 Numerical Factor of Algebraic Fraction (numfactor)
It returns the numerical factor from an algrebraic fraction.
✞
1(%i1) numfactor(15/8 * %e);
✆
The output is
(%o1)
15
8

4.1. ARITHMATIC FUNCTIONS 61
4.1.5 Product of Terms (product)
productis used for product evaluation of a series. Syntax forproductis
✞
1(%i1) product(<expression>, <variable>, <initial value>,<final value>)
✆
Product
n
Y
i=1
i
2
is expanded as
n
Y
i=1
i
2
= 1
2
×2
2
×3
2
×. . .×n
2
If either initial or ﬁnal value is symbol, then output is just symbolic representa-
tion of product. The product of a series of square of ﬁrst ‘n’ integers is evaluated
as
✞
1(%i1) product(x^2, x, a, b)
✆
The output is
(%o1)
b
Y
x=a
x
2
If initial & ﬁnal values are integers then
✞
1(%i1) product(x^2, x, 1, 5)
✆
✞
(\%o1) 14400
✆
4.1.6 Maximum Value (max)
maxreturns the maximum value of supplied arguments as vector or matrix.
✞
1(%i1) max(20,15,47,85,0,-2)
✆
✞
(%o1) 85
✆
4.1.7 Minimum Value (min)
minreturns the minimum value of supplied arguments as vector or matrix.
✞
1(%i1) min(20,15,47,85,0,-2)
✆
✞
(%o1) -2
✆

62 CHAPTER 4. ARITHMETIC
4.1.8 Modulo (mod)
It returns the modulo of two expressions as shown in the syntax ofthis function.
✞
1(%i1) mod(<expression 1>, <expression 2>)
✆
Modulo between two pure integer values is given by
✞
1(%i1) mod(10,3)
✆
✞
(%o1) 1
✆
Modulo for algebraic relations
✞
1(%i1) mod(a^3-b^3,a-b);
✆
(
4.1.9 Square Root (sqrt)
It returns the square root of given number.
✞
1(%i1) numer:true;
(%i2) sqrt(10)
✆
✞
(%o2) 3.16227766016838
✆
4.1.10 Sorting Elements (sort)
4.1.11 Simpliﬁcation Environment (simp)
Whensimpis used as anevﬂagwith a value ‘false’, the simpliﬁcation is sup-
pressed only during the evaluation phase of an expression. The ﬂagcan not
suppress the simpliﬁcation which follows the evaluation phase. The simpliﬁ-
cation is switched oﬀ globally. The expressionsin(1.0)is not simpliﬁed to its
numerical value. Thesimp-ﬂag switches the simpliﬁcation on is
✞
1(%i4) simp:true;
✆
✞
(%o4) true
✆
✞
1(%i5) x:sin(1.0),simp:false;
✆
✞
(%o5) 0.8414709848078965
✆

Chapter 5
Algebra
This section includes the functions can be used in algebra.
5.1 Linear Equations
5.1.1 Algebraic Evaluation (algebraic)
By defaultalgebraicis ‘false’. It must be set ‘true’ in order to simplify the
algebraic numbers.
5.1.2 Solving Linear Equation (solve)
A linear equation can be solved by usingsolvefunction. An algebraic equation,
f(x) = 0 of a variablex, would be arranged in form off(x) = 0 before getting
the roots of the equation. If equation is in form off(x) =athen it should be
arranged asf(x)−a= 0 before supplying it as argument to thesolvefunction.
The synopsis of thesolvefunction is
✞
1(%i1)solve([<linearalgebraic equation>],[<variable>])
✆
Example is
✞
1(%i1)solve([x^2-x+1=0],[x])
✆
The output is
(%o1)
"
x=−
√
3i−1
2
, x=
√
3i+ 1
2
#
If we have to solve more than one function then syntax forsolvebecomes
✞
1(%i11) x+y=1
✆
63

64 CHAPTER 5. ALGEBRA
(%o11)x+y= 1
✞
1(%i12) x-y=2
✆
(%o12)x−y= 2
✞
1(%i13)solve([%o11,%o12])
✆
(%o12)
≤≤
y=−
1
2
, x=
3
2
λλ
5.1.3 Simplify Logs, Exponent & Radicals (radcan)
radcansimpliﬁes expressions, which contain logs, exponentials and radicals.For
a somewhat larger class of expressions,radcanproduces a regular form. Two
equivalent expressions in this class do not necessarily have the sameappearance,
but their diﬀerence can be simpliﬁed byradcanto zero. For example
×
log(x+x
2
)−log(x)
∗
a
[log(1 +x)]
a/2
= [log(x+ 1)]
a/2
Maxima code is
✞
1(%i3)radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
✆
(%o3) log (x+ 1)
a
2
5.1.4 Radical Expansion (radexpand)
By default it is ‘true’. It controls the expansion of radicals. When it is ‘true’
for all, factors are pulled outside to the radicals. For example, whenradexpand
is ‘true’,
√
4x
2
becomes 2xas a coeﬃcient of the expression.
5.1.5 Rational Simpliﬁcation (ratsimp)
It is abbrevation of “rational simpliﬁcation”. It simpliﬁes the expression and all
of its subexpressions, including the arguments to non-rational functions. The

5.1. LINEAR EQUATIONS 65
result is returned as the quotient of two polynomials in a recursive form. The
syntax used is
✞
1(%i3)ratsimp(<expression>, <arg 1>, <arg 2>, ... , <arg n>)
✆
For example
✞
1(%i3)sin(x)/x +exp((1+log(x))^2 -log(x));
✆
(%o3)
sin (x)
x
+
e
(log(x)+1)
2
x
✞
1(%i3)sin(x)/x +exp((1+log(x))^2 -log(x));
(%i4)ratsimp(%);
✆
(%o4)
sin (x) +x
2
e
log(x)
2
+1
x
5.1.6 Rationalized Simpliﬁcation Flag (ratsimpexpons)
By default it is ‘false’. Whenratsimpexponsis true,ratsimpis applied to the
exponents of expressions during simpliﬁcation.
5.1.7 Evaluation of Expression in Environment (ev)
It evals an expression in the environment as speciﬁed by the user in form of
arg1,arg2,. . .,argn. In all of these arguments, at least one must be from the
environmentssimp,noeval,diﬀ,integrate,expand,detout,derivlist,ﬂoat,numer,
pred,evalor other valid function names. The syntax used for this function is
✞
(%i3) ev(<expression>, <arg 1>, <arg 2>, ... , <arg n>)
✆
A functionf(x) =x
2
+2, is to be derivated with respect to independent variable
‘x’ by using theevfunction is
✞
1(%i3) x+1;
(%i4) ev(%,1);
✆
(%o4) 2

66 CHAPTER 5. ALGEBRA
5.1.8 Evaluation of Expression (eval)
It is used as an argument in the functionevto cause an extra evaluation of
expression or relation.
5.1.9 Expansion of Algebraic Relations (expand)
expandexpands to the algebraic relations consisiting product of sum of two
or more independent variables. Exponents of sum of two or more variables or
algebraic constants are splited in simpliﬁed terms and numerical constants are
made corresponding coeﬃcients of their respective terms. The syntax used is
✞
(%i3)expand(<expression>)
✆
For example
✞
1(%i3) (a+b)^2;
(%i4)expand(%);
✆
(%o4)b
2
+ 2ab+a
2
Another form of the syntax ofexpandis
✞
(%i3)expand(<expression>, <mpnum>, <maximum power of denominator>)
✆
Here ‘mpnum’ limits the expansion of those terms whose exponent is less than or
equal to ‘mpnum’ in the numerator. Similarly ‘mpden’ limits the expansion of
those terms whose exponent is less than or equal to ‘mpden’ in the denominator.
This limitation is applicable when power of exponents is positive. It is helpful
to expand the speciﬁc radicals in the rational algebraic expression.For example
✞
1(%i3) (a+b)^2;
(%i4)expand(%, 1);
✆
Does not exapanded as the exponent of numerator is ‘2’ that is greater than
the ‘mpnum’ value ‘1’. Remember if there is negative exponents of a radical
value in rational expressions then their power is made positive by changing their
position from the numerator to denominator and vice versa.
5.2 Inequality & Polynomial
5.2.1 Equality (equal)
equalcompares two expressions, whether two expressions are equal or not. The
stat of comparison is returned when it is asked by using functionis. It is used
for true state. If the condition is ‘true’, then the function is proven by facts and
assumed in database.

5.2. INEQUALITY & POLYNOMIAL 67
✞
(%i3) equal(x, x^2);
2(%i4)is(%);
✆
(%o4)unknown
✞
(%i3) equal(x*x,x^2);
2(%i4)is(%);
✆
(%o4)true
5.2.2 Inequality (notequal)
notequalcompares two expressions, whether two expressions are equal or not.
The stat of comparison is returned when it is asked by using functionis. It is
used for false state.
✞
(%i3) notqual(x, x^2);
2(%i4)is(%);
✆
(%o4)unknown
✞
(%i3)notequal(x*x,x^2);
2(%i4)is(%);
✆
(%o4)false
5.2.3 Solve Algebraic Polynomials (algsys)
algsysused to solve the simultaneous polynomial equations for the list of the
variables. All the polynomial equations and list of variables to be supplied as
arguments to the function must be in vector form. The syntax used is
✞
(%i3) algsys([s^3+1], [s]);
✆

68 CHAPTER 5. ALGEBRA
(%o3)
s=−1
s=
√
3i+ 1
2
s=−
√
3i−1
2
5.2.4 Solve Linear Equations (linsolve)
linsolveis used to solve the simultaneous linear equations for the list of variables.
The expressions must be polynomials in the variables and may be equations.
All the linear equations and list of variables to be supplied as arguments to the
function must be in vector form. The syntax used is
✞
1(%i3)linsolve([2*x+8*y-4*z=10, 2*x+y+4*z=8, x-8*y-z=-20], [x, y, z]);
✆
(%o3)x=−
45
89
y=
198
89
z=
151
89
5.2.5 Check The Stat Of Database (is)
isattempts to determine whether the predicate expression is provable from the
facts in the assume database. If the proven state from the facts is not assumed
then it returns the status ‘unknown’.
✞
1(%i3)notequal(x*x, x^2);
(%i4)is(%);
✆
(%o4)false
5.2.6 Check The Stat Of Database (maybe)
maybesimilar tois, attempts to determine whether the predicate expression
is provable from the facts in the assume database. If state is not assumed in
database, it returns ‘unknown’ result.
✞
(%i3)notequal(x, x^2);
2(%i4)maybe(%);
✆

5.2. INEQUALITY & POLYNOMIAL 69
(%o4)unknown
✞
(%i5)notequal(x*x, x^2);
2(%i6)maybe(%);
✆
(%o6)false
5.2.7 Algebraic Substitution (subst)
This function substitutes old variable by new variable. The syntax for this
function is
✞
(%i3)subst(<new var>, <old var>, <expression>);
✆
For example
✞
1(%i5)subst(y, x, x^n-a^n);
✆
(%o6)y
n
−a
n

70 CHAPTER 5. ALGEBRA

Chapter 6
Logarithm
6.0.8 Expoential (exp)
It represents to the exponential base. The syntax used for thisfunction is
✞
1(%i5)exp(<a constantornumber>)
✆
It is similar to the %e
x
where ‘x’ is a constant or a number. If%enumerset
true then numerical value of %e
x
is placed in place of %e
x
.
6.0.9 Logarithm (log)
6.0.10 Absolute Logarithm (logabs)
6.0.11 Arc Logarithm (logarc)
6.0.12 logconcoeﬀp
6.0.13 logcontract
6.0.14 Logarithmic Expansion (logexpand)
6.0.15 lognegint
6.0.16 Logarithmic Numerical Value (lognumer)
6.0.17 Logarithmic Simpliﬁcation (logsimp)
71

72 CHAPTER 6. LOGARITHM

Chapter 7
Calculus
Following function are useful for calculus.
7.1 Limit
7.1.1 Finding Limit (limit)
Functionlimitis used to ﬁnd a limit of a function at a point either in positive
direction or in negative direction. The syntax used is
✞
1(%i5)limit(
<function>,
3 <independent variable>,
<limitpoint>,
5 <direction>
)
✆
The values of ‘direction’ are “plus” for a limit from above, “minus” fora limit
from below. For example
✞
(%i5)limit(sin(x)/x, x, 0, plus);
✆
✞
(%o5) 1
✆
7.1.2 Limit Substitution (limsubst)
It prevents limit from attempting substitutions on unknown forms.By default
it is ‘false’. The substitution is allowed iflimsubstis set to ‘true’.
7.1.3 Limit By Taylor Series (tlimit)
For using the Taylor series expansion to ﬁnd the limit use functiontlimit. The
syntax of functiontlimitis identical to the functionlimit.
73

74 CHAPTER 7. CALCULUS
✞
1(%i5)tlimit(
<function>,
3 <independent variable>,
<limitpoint>,
5 <direction>
)
✆
For example
✞
(%i5)tlimit(sin(x)/x, x, 0, plus);
✆
✞
(%o5) 1
✆
7.1.4 Taylor Limit Switch (tlimswitch)
By default it is ‘true’. Whentlimswitchis ‘true’, the limit command will use a
Taylor series expansion if the limit of the input expression cannot be computed
directly. Whentlimswitchis ‘false’ and the limit of input expression cannot be
computed directly, limit will return an unevaluated limit expression.
7.2 Derivative
7.2.1 Derivative of a Function (diﬀ)
diﬀis used to get derivative of a function with respect to an independent vari-
able. The syntax used is
✞
1(%i5) diff(
<function>,
3 <independent variable>,
<order of diff>
5 )
✆
If ‘order of diﬀ’ is omitted then only ﬁrst order derivative is calculated. Example
is
✞
1(%i5) diff(sin(x)*cos(x),x)
✆
(%o5) cos
2
(x)−sin
2
(x)
A muti-variable and multi-order diﬀerentiation of a function is performed by
using following syntax.

7.2. DERIVATIVE 75
✞
1(%i5) diff(
<function>,
3 <var 1>,
<order var 1>,
5 <var 2>,
<order var 2>, ...
7 )
✆
Example is
✞
1(%i5)’diff(sin(x)*cos(y),x,2,y,2);
✆
(%o5)
d
4
dx
2
dy
2
[sin(x) cos(y)]
If ‘independent variable’ is not provided then the result contains
✞
1del(<independent variable>)
✆
that represents to the diﬀerential of ‘independent variable’ and return value is
so called “total diﬀerential”. For example
y= cos(x)×sin(x)
Its total derivative is
dy=
−
cos
2
(x)−sin
2
(x)
·
dx
Single variable example in Maxima is
✞
1(%i5) diff(sin(x)*cos(x))
✆
(%o5) (cos
2
(x)−sin
2
(x))del(x)
A multivariable example is
✞
1(%i5) diff(sin(x)*cos(y))
✆
(%o5) cos(x) cos(y)del(x)−sin(x) sin(y)del(y)
diﬀfunction can be used withintegratefunction to evaluate relation for “deriva-
tive of integral” written as
d
dx
Z
b(x)
a(x)
f(t)dt
Maxima example is given like

76 CHAPTER 7. CALCULUS
✞
1(%i5) diff(
integrate(
3 cos(t),
t,
5 0,
x^2
7 ),
x,
9 1
);
✆
(%o5) 2xcos(x
2
)
7.2.2 Partial Drivative (gradef)
7.2.3 Total Diﬀerential (del)
If ‘independent variable’ is not provided then the result containsdel(¡independent
variable¿)that represents to the diﬀerential of ‘independent variable’ and return
value is so called “total diﬀerential”. Single variable example is
✞
(%i5) diff(sin(x)*cos(x))
✆
(%o5) (cos
2
(x)−sin
2
(x))del(x)
A multivariable example is
✞
1(%i5) diff(sin(x)*cos(y))
✆
(%o5) cos(x) cos(y)del(x)−sin(x) sin(y)del(y)
A functional deﬀerentiation can be obtained by using functiondiﬀlike the
syntax.
✞
1(%i5) depends([f,g],u);
(%i5) diff(f*g,u);
✆
(%o5)f

d
d u
g

+

d
d u
f

g

7.2. DERIVATIVE 77
7.2.4 Dependence of Function (depends)
Supposefandgare two independent functions and they are dependend of
uthen the function will be likef(u) andg(u). Here functionsfandgare
dependent to variableu. To deﬁne the dependency of function to independent
variable,dependsfunction is used like the syntax
✞
(%i5) depends(<function>, <dependent variable>)
✆
Functions may be dependent to one or more variables. The syntax will be
modiﬁed like
✞
1(%i5) depends(
[<function 1>,<function 2>],
3 [<dependent var 1>, <dependent var 2>]
)
✆
Here functions and independent variables are in vector form. A combination
of vector and scalar for functions and independent variables can also be used.
Single function and single variable xample is
✞
(%i5) depends (f, x);
✆
(%o5) [ f(x)]
Multi-functions and multi-variables example is
✞
1(%i5) depends ([r, s], [u, v, w]);
✆
(%o5) [r(u, v, w), s(u, v, w)]
7.2.5 Show Dependencies (dependencies)
It returns the previously deﬁned dependent functions. The syntax used is
✞
1(%i5) dependencies;
✆
7.2.6 Notation of Derivative (derivabbrev)
Whenderivabbrevis ‘true’, symbolic derivatives are displayed as subscripts.
Otherwise, derivatives are displayed in the Leibniz notationdy/dx.
✞
1(%i4) derivabbrev:true;
(%i5) diff(f(x),x)
✆

78 CHAPTER 7. CALCULUS
(%o5)f(x)x
7.2.7 Value of Function at a Point (atvalue)
atvalueassigned a value to function when condition is met. The syntax used is
✞
(%i5) atvalue(<function>, <conditions>, <rep value>)
✆
This function is used to assign boundary values to a function. For example,
whenx= 0 andy= 0 then assign functionf(x, y) value ‘a’. The syntax is
✞
1(%i5) atvalue(f(x,y), [x=0,y=0], a)
✆
(%o5)a
If ‘rep value’ is not a constant value and it is expression consisting independent
variables ‘x’ and ‘y’ then ﬁrst independent variable of function is represented
as ‘@1’ and second indpenedent variable is represented as ‘@2’ and so on. For
example
✞
1(%i5) atvalue(f(x,y), [x=0,y=0], x+y)%firstvar=x &secondvar=y
✆
(%o5) @2 + @1
Example of three variables function with two variables condition and replace-
ment value.
✞
1(%i5) atvalue(f(x,z,y), [x=0,y=0], x+y) %firstvar=x &thirdvar=y
✆
(%o5) @3 + @1
7.2.8 Degree of Derivative (derivdegree)
Returns the highest degree of derivative of dependent variableywith respect to
the independent variablexoccuring in expression. The syntax used is
✞
1(%i5) atvalue(<expression>, <dependent var>, <independent var>)
✆
Here ‘expression’ may be a function name or a diﬀerential equation.

7.3. INTEGRATION 79
7.2.9 Partial Derivative (express)
expressexpands the diﬀerential operators in terms of partial derivatives. It ex-
pands the diﬀerential operatorsgrad,div,curl,laplacianand the cross product.
The syntax used is
✞
1(%i5)express(<expression>)
✆
7.2.10 Selective Derivative (derivlist)
It causes only diﬀerentiations with respect to the indicated variables, within the
evcommand.
✞
1(%i5) derivlist(var_1, ...., var_k)
✆
7.2.11 Substitution in Derivatives (derivsubst)
Whenderivsubstis ‘true’, a non-syntactic substitution such as
✞
1(%i5)subst(x,’diff (y, t), ’diff (y, t, 2))
✆
yields
✞
1(%i6)’diff(x, t)
✆
By default it is ‘false’.
7.3 Integration
7.3.1 Anti Derivative of a Function (integrate)
integratefunction is used to integrate a function. Integration of functionf(x)
within lower and upper limit is expressed as
I=
Z
u
l
f(x)dx
The syntax used in integration of a function is
✞
1(%i5)integrate(<function>, <base of int>, <lowerlimit>, <upperlimit>)
✆
If ‘lower limit’ and ‘upper limit’ are omitted then function is integrated inthe
domain of−∞to +∞. Example is
✞
1(%i5)integrate(sin(x)*cos(x),x)
✆

80 CHAPTER 7. CALCULUS
(%o5) −0.5 cos
2
(x)
If limits are supplied then
✞
1(%i5)integrate(sin(x)*cos(x),x,1,2)
✆
✞
(%o5) 0.059374196079117
✆
7.3.2 Changing The Variable (changevar)
Let a function is given byf(x) wherexis independent variable and function is
dependent on thex. Now the independent variable,xcan be substituted by a
new variabley, given as functionf(x, y) = 0, using this function like.
✞
1(%i3) changevar(<expr>, <f(x,y)>, <new var y>, <old var x>)
✆
For example, substitution of old variablexwith new variableyaccording to the
relationy=x+ 1 in integral
Z
1
0
√
x+ 1−1
x
dx
is
✞
1(%i3)’integrate((\sqrt(x+1)-1)/x,x,0,1)$
(%i4) changevar(%, -y+x+1, y, x);
✆
Output after substituting the variable, new integration form will be
(%o4)
Z
2
1
1
√
y+ 1
dy
7.3.3 Double Integration (dblint)
Double integration of a two variable functionf(x, y) is given by
Idbl=
Z
ux
lx
Z
u(x)
l(x)
f(x, y)dy dx
Double integration is area covered by a function and a line within the ﬁxed
boundary limits. The syntax for double integration in Maxima is
✞
(%i5) dblint(<function of x & y>, <l(x)>, <u(x)>, <lx>, <ux>)
✆

7.3. INTEGRATION 81
Here,l(x) is lower limit ofyas a function ofxandu(x) is upper limit ofy
as a function ofx. Similarlylxis lower limit ofxanduxis upper limit ofx.
Example is
✞
1(%i5) f(x,y):=x*y;
(%i6) l(x):=1.0;
3(%i7) u(x):=x;
(%i8) dblint(’f,’l,’u,0.0,1.0);
✆
✞
(%o8) -0.125
✆

82 CHAPTER 7. CALCULUS

Chapter 8
Ordinary Diﬀerential
Equation
An ordinary diﬀerential equation (ODE) is an equality involving a function and
its derivatives. An ODE of ordernis an equation of the form
F

x, y, y
′
, ..., y
(n)

= 0
whereyis a function ofx,y
′
=dy/dxis the ﬁrst derivative ofywith respect to
x, andy
(n)
=d
n
y/dx
n
is then
th
derivative ofywith respect tox. Nonhomo-
geneous ordinary diﬀerential equations can be solved if the general solution to
the homogenous version is known, in which case the undetermined coeﬃcients
method or variation of parameters can be used to ﬁnd the particular solution.
Many ordinary diﬀerential equations can be solved exactly in Maxima using
dsolvefunction, and numerically by enablingnumervariable. An ODE of order
nis said to be linear if it is of the form
an(x)y
(n)
+a
(n−1)(x)y
(n−1)
+...+a1(x)y
′
+a0(x)y=Q(x)
A linear ODE whereQ(x) = 0 is said to be homogeneous.
8.1 Boundary Value Problems
8.1.1 Boundary Value Problem (bc2)
bc2is acronym of boundary value problem for second order diﬀerentialequation.
It receives ﬁve arguments. The syntax used forbc2is
✞
1(%i5) bc2(<solution>, <x_1>, <y_1>, <x_2>, <y_2>)
✆
First argument ‘solution’ is the solution of second order diﬀerentialequation.
It has two constants those are obtained by substituting the initialand ﬁnal
boundary values (x1, y1) and (x2, y2) respectively. For example
83

84 CHAPTER 8. ORDINARY DIFFERENTIAL EQUATION
✞
1(%i5) bc2(y=x+%k1+%k2, x=0, y=1, x=1, y=2);
✆
(%o5)y=x+ 1
8.1.2 Linear Ordinary Diﬀerential Equations (desolve)
The functiondesolvesolves systems of linear ordinary diﬀerential equations
using Laplace transform. The syntax used is
✞
1(%i5) desolve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])
✆
For example
✞
1(%i5)
✆
(%o5)
8.1.3 Initial Value Problem of First Order (ic1)
A ﬁrst-order ordinary diﬀerential equation is either in form of
dy
dx
=F(x, y)
or in form of
dy
dx
+p(x)y=q(x)
and its solution is
y=
R
h
e
R
p(x)dx
i
q(x)dx+c
e
R
p(x)dx
To get the constant value ‘c’, we substitute the initial values in the solution of
ﬁrst order diﬀerential equation. The value of ‘c’ is again put into thesolution
of ﬁrst order equation and ﬁnally it gives the ﬁnal solution. The functionic1
solves initial value problem of ﬁrst order diﬀerential equation. It accepts three
arguments. ‘solution’ is the solution of ﬁrst order diﬀerential equation. It has
one constant whose value is obtained by substituting initial boundary values
(x1, y1). The syntax used is
✞
1(%i5)ic1(<solution>, <x_1>, <y_1>)
✆
For example

8.2. ODE 85
8.1.4 Initial Value Problem of Second Order (ic2)
A second-order linear ODE with variable coeﬃcients
d
2
y
dx
2
+p(x)
dy
dx
+q(x)y= 0
Solution of this equation is
?
The functionic2solves initial value problem of second order diﬀerential equa-
tion. It accepts four arguments. ‘solution’ is the solution of second order diﬀer-
ential equation. It has one constant,k, and one derivative of dependent variable
with respect to independent variable,dy/dx, after ﬁrst anti-derivative of second
order diﬀerential equation. Before performing second anti-derivative to get ﬁnal
solution of second order diﬀerential equation, constantkis obtained by sub-
stituting the initial boundary values and derivative value, [dy/dx]
x1,y1
at the
initial boundary value in the ‘solution’. The syntax used is
✞
1(%i5)ic2(<solution>, <x_1>, <y_1>, <deriv x_1 y_1>)
✆
For example
8.2 ODE
8.2.1 Ordinary Deferential Equation of Second Order (ode2)
The functionode2solves an ordinary diﬀerential equation (ODE) of ﬁrst or
second order. It receives three arguments, ‘solution’, ‘dependent variable’ and
‘independent variable’. The syntax used is
✞
1(%i5)ode2(<equation>, <dependent variable>, <independent variable>)
✆
For example
✞
1(%i5)ode2(’diff(y,x) -y=1, y,x);
✆
(%o5)y=
−
%c−e
−x
·
e
x

86 CHAPTER 8. ORDINARY DIFFERENTIAL EQUATION

Chapter 9
Trigonometric
Following trigonometric functions and relations are useful for mathematical so-
lutions.
9.1 Trigonometric Functions
9.1.1 Sine (sin)
sinis acronym of “sine”. It computes the sine of supplied argument in radian.
The principal domain of sine is−
π
2
≤x≤
π
2
. General domain of sine is all real
numbers. If argument is outside the principal domain then Maxima computes
nothing. Sine function is given by
y= sin(x)
Herexis argument/angle and measured in radian.
✞
1(%i5)sin(10)
✆
✞
(%o5)sin(10)
✆
✞
1(%i5)sin(3.14)
✆
✞
(%o5) 0.0015926529164868
✆
9.1.2 Cosine (cos)
cosis acronym of “cosine”. It computes the cosine of supplied argument in
radian. The principal domain of cosine is 0≤x≤π. General domain of cosine
is all real numbers. If supplied argument is outside the domain then Maxima
computes nothing.
87

88 CHAPTER 9. TRIGONOMETRIC
✞
1(%i5) cos(10)
✆
✞
(%o5) cos(10)
✆
✞
1(%i5) cos(0)
✆
(%o5) 1
9.1.3 Tangent (tan)
tanis acronym of “tangent”. It computes the tangent of supplied argument in
radian. The principal domain of tangent is−
π
2
< x <
π
2
. General domain of
tangent is all real numbers except the odd integer multiple ofπ/2. If supplied
argument is outside the domain then Maxima computes nothing.
✞
1(%i5)tan(10)
✆
✞
(%o5)tan(10)
✆
✞
1(%i5)tan(3.14)
✆
(%o5) −0.0015926549364072
9.1.4 Cotangent (cot)
cotis acronym of “cotangent”. It computes the cotangent of supplied argument
in radian. The principal domain of secant is 0< x < π. General domain
of cotangent is all real numbers except the integer multiple ofπ. If supplied
argument is outside the domain then Maxima computes nothing.
✞
1(%i5) cot(10)
✆
✞
(%o5) cot(10)
✆
✞
1(%i5) cot(%pi/2)
✆
(%o5) 0

9.2. HYPERBOLIC TRIGONOMETRIC FUNCTIONS 89
9.1.5 Secant (sec)
secis acronym of “secant”. It computes the secant of supplied argument in
radian. The principal domain of secant is 0≤x≤π, x6=
π
2
. General domain
of secant is all real numbers except the odd integer multiple ofπ/2. If supplied
argument is outside the domain then Maxima computes nothing.
✞
1(%i5)sec(10)
✆
✞
(%o5)sec(10)
✆
✞
1(%i5)sec(0)
✆
(%o5) 1
9.1.6 Cosecant (cosec)
cosecis acronym of “cosecant”. It computes the cosecant of supplied argument
in radian. The principal domain of cosecant is−
π
2
≤x≤
π
2
, x6= 0. General
domain of cosecant is all real numbers except the integer multiple ofπ. If
supplied argument is outside the domain then Maxima computes nothing.
✞
1(%i5) csc(10)
✆
✞
(%o5) csc(10)
✆
✞
1(%i5) csc(%pi/2)
✆
(%o5) 1
9.2 Hyperbolic Trigonometric Functions
9.2.1 Hyperbolic Sine (sinh)
sinis acronym of ‘hyperbolic sine”. It computes the hyperbolic sine of supplied
argument in radian. The principal domain of hyperbolic sine isR. If supplied
number is outside the domain then Maxima computes nothing.
✞
1(%i5)sinh(10)
✆

90 CHAPTER 9. TRIGONOMETRIC
✞
(%o5)sinh(10)
✆
✞
1(%i5)sinh(3.14)
✆
✞
(%o5) 11.53029203041011
✆
9.2.2 Hyperbolic Cosine (cosh)
coshis acronym of “hyperbolic cosine”. It computes the hyperbolic cosine of
supplied argument in radian. The principal domain of hyperbolic cosineisR.
If supplied argument is outside the domain then Maxima computes nothing.
✞
1(%i5) cosh(10)
✆
✞
(%o5) cosh(10)
✆
✞
1(%i5) cosh(0)
✆
(%o5) 1
9.2.3 Hyperbolic Tangent (tanh)
tanhis acronym of “hyperbolic tangent”. It computes the hyperbolic tangent
of supplied argument in radian. The principal domain of hyperbolic tangent is
R. If supplied argument is outside the domain then Maxima computes nothing.
✞
1(%i5)tanh(10)
✆
✞
(%o5)tanh(10)
✆
✞
1(%i5)tanh(3.14)
✆
(%o5) 0.99626020494583
9.2.4 Hyperbolic Cotangent (coth)
The principal domain of hyperbolic cotangent isR− {0}.

9.3. INVERSE TRIGONOMETRIC FUNCTIONS 91
9.2.5 Hyperbolic Secant (sech)
sechis acronym of “hyperbolic secant”. It computes the hyperbolic secant of
supplied argument in radian. The principal domain of hyperbolic secant isR.
If supplied argument is outside the domain then Maxima computes nothing.
✞
1(%i5)sech(10)
✆
✞
(%o5)sech(10)
✆
✞
1(%i5)sech(0)
✆
(%o5) 1
9.2.6 Hyperbolic Cosecant (csch)
The principal domain of hyperbolic cosecant isR− {0}.
9.3 Inverse Trigonometric Functions
9.3.1 Inverse Sine (asin)
asinis acronym of “inverse of sine”. It computes the sine inverse of supplied
number. The principal domain of inverse sine is [−1,1]. If supplied number is
outside the domain then Maxima computes nothing.
✞
1(%i5) asin(10)
✆
✞
(%o5) asin(10)
✆
✞
1(%i5) asin(1)
✆
(%o5)
π
2
9.3.2 Inverse Cosine (acos)
acosis acronym of “inverse of cosine”. It computes the cosine inverse of supplied
number. The principal domain of inverse cosine is [−1,1]. If supplied number
is outside the domain then Maxima computes nothing.

92 CHAPTER 9. TRIGONOMETRIC
✞
1(%i5) acos(10)
✆
✞
(%o5) acos(10)
✆
✞
1(%i5) acos(0)
✆
(%o5)
π
2
9.3.3 Inverse Tangent (atan)
atanis acronym of “inverse of tangent”. It computes the tangent inverse of
supplied number. The principal domain of inverse tangent isR. If supplied
number is outside the domain then Maxima computes nothing.
✞
1(%i5) atan(10)
✆
✞
(%o5) atan(10)
✆
✞
1(%i5) atan(3.14)
✆
(%o5) 1.262480664599468
9.3.4 Inverse Tangent Type Two (atan2)
atan2yields the value ofatan(y/x) in the interval−πtoπ. If supplied number
is outside the domain then Maxima computes nothing.
9.3.5 Inverse Cotangent (acot)
acotis acronym of “inverse of cotangent”. It computes the cotangent inverse of
supplied number. The principal domain of inverse cotangent isR. If supplied
number is outside the domain then Maxima computes nothing.
✞
1(%i5) acot(10)
✆
✞
(%o5) acot(10)
✆
✞
1(%i5) acot(0.5)
✆

9.4. INVERSE HYPERBOLIC TRIGONOMETRIC FUNCTIONS 93
(%o5) 1.10714871779409
9.3.6 Inverse Secant (asec)
asecis acronym of “inverse of secant”. It computes the secant inverse of supplied
number. Theprincipal domain of inverse secant isR−(−1,1). If supplied
number is outside the domain then Maxima computes nothing.
✞
1(%i5) asec(10)
✆
✞
(%o5) asec(10)
✆
✞
1(%i5) asec(1)
✆
(%o5) 2.993222846126381i
9.3.7 Inverse Cosecant (acsc)
acscis acronym of “inverse of cosecant”. It computes the cosecant inverse of
supplied number. The principal domain of inverse cosecant isR−(−1,1). If
supplied number is outside the domain then Maxima computes nothing.
✞
1(%i5) acsc(10)
✆
✞
(%o5) acsc(10)
✆
✞
1(%i5) acsc(1)
✆
(%o5)
π
2
9.4 Inverse Hyperbolic Trigonometric Functions
9.4.1 Inverse Hyperbolic Sine (asinh)
asinhis acronym of “inverse of hyperbolic sine”. It computes the hyperbolic
sine inverse of supplied number. The principal domain of inverse hyperbolic
sine is−∞< x <∞. If supplied number is outside the domain then Maxima
computes nothing.

94 CHAPTER 9. TRIGONOMETRIC
✞
1(%i5) asinh(10)
✆
✞
(%o5) asinh(10)
✆
✞
1(%i5) asinh(0.1)
✆
(%o5) 0.099834078899208
9.4.2 Inverse Hyperbolic Cosine (acosh)
acoshis acronym of “inverse of hyperbolic cosine”. It computes the cosine hy-
perbolic inverse of supplied number. The principal domain of inverse hyperbolic
cosine is 1≤x <∞. If supplied number is outside the domain and zero then
Maxima computes nothing.
✞
1(%i5) acosh(0)
✆
✞
(%o5) acosh(0)
✆
✞
1(%i5) acosh(0.5)
✆
(%o5) 1.047197551196598i−1.1102230246251565×10
−16
9.4.3 Inverse Hyperbolic Tangent (atanh)
atanhis acronym of “inverse of hyperbolic tangent”. It computes the hyperbolic
tangent inverse of supplied number. The principal domain of inversehyperbolic
tangent is−1< x <1. If supplied number is outside the domain then Maxima
computes nothing.
✞
1(%i5) atanh(10)
✆
✞
(%o5) atanh(10)
✆
✞
1(%i5) atanh(3.14)
✆
(%o5) 0.32994497940173−1.570796326794897i

9.4. INVERSE HYPERBOLIC TRIGONOMETRIC FUNCTIONS 95
9.4.4 Inverse Hyperbolic Cotangent (acoth)
acoshis acronym of “inverse of hyperbolic cotangent”. It computes thecotan-
gent hyperbolic inverse of supplied number. Theprincipal domain of inverse
hyperbolic cotangent is{−∞< x <−1} ∪ {1< x <∞}. If supplied number is
outside the domain and zero then Maxima computes nothing.
✞
1(%i5) acoth(0)
✆
✞
(%o5) acoth(0)
✆
✞
1(%i5) acoth(0.5)
✆
(%o5) 0.54930614433405−1.570796326794897i
9.4.5 Inverse Hyperbolic Secant (asech)
asechis acronym of “inverse of hyperbolic secant”. It computes the hyperbolic
secant inverse of supplied number. The principal domain of inverse hyperbolic
secant is 0< x≤1. If supplied number is outside the domain then Maxima
computes nothing.
✞
1(%i5) asech(10)
✆
✞
(%o5) asech(10)
✆
✞
1(%i5) asech(0.1)
✆
(%o5) 2.993222846126381
9.4.6 Inverse Hyperbolic Cosecant (acsch)
acschis acronym of “inverse of hyperbolic cosecant”. It computes the cose-
cant hyperbolic inverse of supplied number. The principal domain of inverse
hyperbolic cosecant is−∞< x <∞, x6= 0. If supplied number is outside the
domain and zero then Maxima computes nothing.
✞
1(%i5) acsch(0)
✆
✞
(%o5) acsch(0)
✆

96 CHAPTER 9. TRIGONOMETRIC
✞
1(%i5) acsch(0.9)
✆
(%o5) 0.95780044920067
9.5 Trigonometric Identities
9.5.1 Half Angle (halfangles)
The default value of variablehalfanglesis ‘false’. If it is set to true then trigono-
metric functions having half angles (x/2) as arguments are expands in form of
angle (x) arguments.
✞
1(%i5)halfangles;
(%i6)sin(x/2);
✆
(%o6) sin(x/2)
✞
(%i5)halfangles:true;
2(%i6)sin(x/2);
✆
(%o6)
(−1)
ﬂoor(
x
2π)
√
1−cosx
√
2
9.5.2 Expansion of Trigonometric Functions (trigexpand)
Iftrigexpandis true then trigonometric expressions are expanded in the form of
sin and cos or %e. This function can be used either in form of ﬂag or in form
of function. In ﬂag form, the default value oftrigexpandis ‘false’. For example
✞
(%i5)trigexpand:true;
2(%i6)sin(3*x);
✆
(%o6) 3 cos
2
(x) sin(x)−sin
3
(x)
By using function directly like

9.5. TRIGONOMETRIC IDENTITIES 97
✞
(%i5)trigexpand(sin(3*x));
✆
(%o5) 3 cos
2
(x) sin(x)−sin
3
(x)
9.5.3 Sum of Arguments (trigexpandplus)
By defaulttrigexpandplusis set to ‘true’. If it is ‘true’ then expansion for
trigonometirc operator for “summation” of arguments is take place. If it is
‘false’ thentrigexpanddoes not expand the trigonometric expression having
arguments in “sum” form. For example
✞
1(%i4)trigexpandplus:false;
(%i5)trigexpand(sin(x+y));
✆
(%o5) sin(y+x)
Iftrigexpandplusis set to ‘true’ or by default it is ‘true’ then
✞
(%i4)trigexpandplus:true;
2(%i5)trigexpand(sin(x+y));
✆
(%o5) cos(x) sin(y) + sin(x) cos(y)
9.5.4 Product of Arguments (trigexpandtimes)
By defaulttrigexpandtimesis set to ‘true’. If it is ‘true’ then expansion for
trigonometirc opertors for “product” of arguments is take place. If it is ‘false’
thentrigexpanddoes not expand the trigonometric expressions. For example
✞
(%i4)trigexpandtimes:false;
2(%i5)trigexpand(sin(3*y));
✆
(%o5) sin(3y)
Iftrigexpandtimesis set to ‘true’ or by default it is ‘true’ then
✞
(%i4)trigexpandtimes:true;
2(%i5)trigexpand(sin(3*y));
✆

98 CHAPTER 9. TRIGONOMETRIC
(%o5) 3 cos
2
(y) sin(y)−sin
3
(y)
9.5.5 Inverse Sequence Flag (triginverses)
triginversescontrols the sequence of function and arcfunction in simpliﬁcation
of trigonometric inverse function. It is ﬂag function. By default itsvalue is
‘true’. It has three values.
1. Iftriginversesis set to “all” then trigonometric operation in sequence
sin(sin
−1
(x))≈sin(arcsin(x))
and
sin
−1
(sin(x))≈arcsin(sin(x))
simpliﬁed tox. The same is applicable to all trigonometric operators (cos,
tan, cot, sec and csc).
2. Iftriginversesset to ‘true’ then sequence like
sin
−1
(sin(x))≈arcsin(sin(x))
is set turned oﬀ.
3. Iftriginversesset to ‘false’ then both sequences
sin(sin
−1
(x))≈sin(arcsin(x))
and
sin
−1
(sin(x))≈arcsin(sin(x))
are set turned oﬀ.
9.5.6 Expansion of Rational Fractions (trigrat)
This function performs canonical simpliﬁed of a trigonometrical expression. The
expression is a rational fraction of several sine, cosine or tangent. These argu-
ments are in linear form in some variables andπ/n, whereninteger. The
result is a simpliﬁed fraction with linear numerator and denominator in sine
and cosine.
✞
(%i5)trigrat(sin(3*a)/sin(a+%pi/3));
✆
(%o5) −sin
2
(a) + 2
√
3 cos(a) sin(a) + cos
2
(a)−1

9.5. TRIGONOMETRIC IDENTITIES 99
9.5.7 Reduce Trigonometric Expressions (trigreduce)
It combines the products and powers of trigonometric and hyperbolic ‘sine’ and
‘cosine’ of argument ‘x’ into those of multiplies of ‘x’. It also tries to eliminate
these functions when they occur in denominators. If ‘x’ is omitted then all
variables in trigonometric relations or expressions are used. The syntax used is
✞
1(%i5)trigreduce(<expression of x>, <variable x>)
✆
or
✞
1(%i6)trigreduce(<multi-variables expression>)
✆
For example
✞
1(%i5)trigreduce(-sin(x)^2+2*cos(x)^2+x);
✆
(%o5)
cos(2x)
2
+ 2

cos(2x)
2
+
1
2

+x−
1
2
9.5.8 Negative Arguments Simpliﬁcation (trigsign)
Iftrigsingis ‘true’ (by default it is ‘true’) then trigonometric functions like
sin(−x) becomes−sin(x). In otherwords it controls the expansion of trigono-
metric operators having negative arguments. It is ﬂag operator.
9.5.9 Trigonometric Simpliﬁcation (trigsimp)
This function includes the trigonometric identity
cos
2
(x) + sin
2
(x) = 1
and
cosh
2
(x)−sinh
2
(x) = 1
to further reduce the result obtained from the application of function liketri-
greduce,ratsimp, andradcanetc. The syntax for the function is
✞
1(%i6)trigsimp(<trigonometric expression>)
✆
9.5.10 Demoivre Expansion (demoivre)
The functiondemoivreconverts one expression without setting the global vari-
abledemoivre. When the variabledemoivreis ‘true’, complex exponentials are
converted into equivalent expressions in terms of circular functions. Ifdemoivre
is set ‘false’ then

100 CHAPTER 9. TRIGONOMETRIC
✞
1(%i6)exp(a + b*%i)
✆
(%o6)e
(i∗b+a)
Ifdemoivreis set ‘true’ then
✞
1(%i5) demoivre:true;
(%i6)exp(a + b*%i)
✆
(%o5)e
a
(isin(b) + cos(b))
The default value ofdemoivreis false.

Chapter 10
Matrix
Following functions are useful for matrices.
10.1 Mathematical Operaton
10.1.1 Matrix (matrix)
The functionmatrixconstructs a matrix of orderm×nwheremis number of
rows andnis number of columns. The syntax for functionmatrixis
✞
(%io) A:matrix([<row1>], [<row2>], ..., [<rown>]);
✆
For example
✞
1(%i1) A:matrix([1,2,3],[2,1,3],[4,7,4]);
✆
(%o1)


1 2 3
2 1 3
4 7 4


10.1.2 Scalar Addition of Matrices (+)
The operator+adds two matrices element by element wise. IfA=Pijand
B=Qijare two matrices then the addition of two matrices will be
Cij=A+B=Pij+Qij
In addition of two matrices corresponding elements are added together and
output is a matrix. Addition operation is performed only if two matrices are of
same order.
101

102 CHAPTER 10. MATRIX
✞
1(%i1) A:matrix([1,2,3],[2,1,3],[4,7,4]);
(%i2) B:matrix([1,2,3],[2,1,3],[4,7,4]);
3(%i3) A+B;
✆
(%o3)


2 4 6
4 2 6
8 14 8


10.1.3 Scalar Subtraction of Matrices (-)
The operator-subtract two matrices element by element wise. IfA=Pijand
B=Qijare two matrices then the subtraction of matrixBfromAwill be
Cij=A−B=Pij−Qij
In subtraction of matrixBfrom matrixA, corresponding elements are sub-
tracted with each other and result is a matrix. Subtraction operation is per-
formed only if two matrices are of same order.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) B:matrix([1,2,3],[2,1,3],[4,7,4]);
3(%i3) A-B;
✆
(%o3)


1 0 0
0 3 0
0 0 0


10.1.4 Scalar Multiplication of Matrices (*)
The operator*multiply two matrices element by element wise. IfA=Pijand
B=Qijare two matrices then the multiplication of matrixBtoAwill be
Cij=A∗B=Pij∗Qij
In scalar multiplication of matrixBto matrixA, corresponding elements are
multiply with each other and result is a matrix. Multiplication operation is
performed only if two matrices are of same order.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) B:matrix([1,2,3],[2,1,3],[4,7,4]);
3(%i3) A*B;
✆

10.1. MATHEMATICAL OPERATON 103
(%o3)


2 4 9
4 4 9
16 49 16


10.1.5 Scalar Division of Matrix (/)
The operator/divides two matrices element by element wise. IfA=Pijand
B=Qijare two matrices then the division of matrixAby matrixBwill be
Cij=A/B=Pij/Qij
In scalar division of matrixAby matrixBis performed by dividing elements of
Aby corresponding elements ofB. Division operation is performed only if two
matrices are of same order.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) B:matrix([1,2,3],[2,1,3],[4,7,4]);
3(%i3) A/B;
✆
(%o3)


2 1 1
1 4 1
1 1 1


10.1.6 Exponent of Matrix (ˆ)
The operatorˆraises power of element of one matrix to the base of another
matrix element by element wise. IfA=PijandB=Qijare two matrices then
the exponentA
B
will be
Cij=A
B
=Pij
Qij
Exponent operation is performed only if two matrices are of same order.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) B:matrix([1,2,3],[2,1,3],[4,7,4]);
3(%i3) A^B;
✆
(%o3)


2 2 3
2 4 3
4 7 4






1 2 3
2 1 3
4 7 4





104 CHAPTER 10. MATRIX
If exponent power is not a matrix but is a number then all elements ofbase
matrix are raised to the power by the number.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) A^2;
✆
(%o3)


4 4 9
4 16 9
16 49 16


If exponent operator is ˆˆ then exponent is noncommutative matrix exponenti-
ation. A scalar base to a matrix power is carried out element by element.
10.1.7 Matrix Dot Product (.)
The dot operator,is for non-commutative multiplication of two matrices. When
“.” is preceded and followed by spaces e.g.A·B, then it distinguishes plainly
from a decimal point in a ﬂoating point number. If two vectors are
~
A= 1ˆi+
2ˆj+ 3
ˆ
kand
~
B= 3ˆi+ 4ˆj+ 5
ˆ
kthenA·Bis given by
A·B= (1ˆi+ 2ˆj+ 3
ˆ
k)·(3ˆi+ 4ˆj+ 5
ˆ
k)
On simpliﬁcation
A·B= 26
In dot product, both lists or vectors or non-commutative matrices should be of
same order. An example is
✞
(%i5) a:[1,2,3];
2(%i6) b:[3,4,5];
(%i7) a . b;
✆
(%o7) 26
Another example
✞
1(%i5) a:[1,2,3];
(%i6) b:[8,9];
3(%i7) a . b;
✆
✞
(%o7) MULTIPLYMATRICES:attempt to multiply nonconformable matrices.
✆

10.2. INVERSE OF MATRIX 105
10.2 Inverse of Matrix
10.2.1 Invert of Matrix Elements (Aˆ-1)
If exponent power is−1 over a matrix then elementwise inversion of elements
of the matrix are take place. IfAijis an element of a matrixAthenA
−1
is a
matrix of the elements 1/Aij.
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2) A^-1;
✆
(%o3)


1/2 1/2 1/3
1/2 1/4 1/3
1/4 1/7 1/4


10.2.2 Inverse of a Matrix (Aˆˆ-1)
Noncommutative exponential ˆˆ is followed by ‘−1’ to a matrix is matrix inver-
sion. IfAis a square matrix then Aˆˆ-1 is matrix inverse.
✞
(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
2(%i2) A^^-1;
✆
(%o3)


5/8−13/8 3/4
−1/2 1/2 0
1/4 3/4−1/2


10.2.3 Append Columns (addcol)
addcolappends one or more lists or matrix,Bp×q, in given matrixAm×nas
columns of matrixAm×n. Row size of matrixBp×qiepshould be equal to the
row size of matrixAm×niem.
✞
(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
2(%i6) B:matrix([2,1,3],[3,1,2],[3,4,4]);
(%i7) addcol(A,B)
✆
(%o7)


1 2 3 2 1 3
4 1 2 3 1 2
5 4 2 3 4 4



106 CHAPTER 10. MATRIX
10.2.4 Append Rows (addrow)
addrowappends one or more lists or matrix,Bp×q, in given matrixAm×nas
rows of matrixAm×n. Column size of matrixBp×qieqshould be equal to the
column size of matrixAm×nien.
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
(%i6) B:matrix([2,1,3],[3,1,2],[3,4,4]);
3(%i7) addrow(A,B)
✆
(%o7)








1 2 3
4 1 2
5 4 2
2 1 3
3 1 2
3 4 4








10.2.5 Ajoint of Matrix (adjoint)
adjointreturns the adjoint of a matrixA. Theadjointmatrix is the transpose
of the matrix of co-factors ofA. Co-factor about an elementAijis the (−1)
i+j
times determinant of sub-matrix found by removingi
th
row andj
th
column.
For obtaining adjoint matrix, the matrix must be a square matrix.
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
(%i6) adjoint(A)
✆
(%o6)


−6 8 1
2−13 10
11 6 −7


10.2.6 Augmented Matrix (augcoefmatrix)
augcoefmatrixreturns the augmented coeﬃcient matrix for the variablesx1,x2,
. . .,xnof the system of linear equationseqn1,. . .,eqnm. This is the coeﬃ-
cient matrix with a column adjoined for the constant terms in each equation.
In otherwords, each equation forms one row of the matrix and its coeﬃcients
construct columns for that row. The syntax is
✞
(%i6) augcoefmatrix(
2 <equaion lists>,
[<firstvariable>, <secondvariable>]
4 );
✆

10.2. INVERSE OF MATRIX 107
✞
(%i5) A:[2*x + 3*y=4, -x + 6*y=10]$
2(%i6) augcoefmatrix(A, [x,y]);
✆
(%o6)

2 3 −4
−1 6−10

10.2.7 Cauchy Matrix (cauchymatrix)
cauchymatrixreturns an×mCauchy matrix with the elements
a[i, j] = 1/(xi+yi)
The second argument ofcauchymatrixis optional. In this case Cauchy matrix
with the elements
a[i, j] = 1/(xi+xi)
The syntax for Cauchy matrix is
✞
(%i6) cauchy_matrix([x_1, x_2, ..., x_m],[y_1, y_2, ..., y_n]);
✆
or
✞
1(%i6) cauchy_matrix([x_1, x_2, ..., x_n]);
✆
For example, With two input arguments for Cauchy matrix
✞
1(%i5) cauchy_matrix([a,b],[c,d])
✆
(%o5)



1
c+a
1
d+a
1
c+b
1
d+b



A single argument Cauchy matrix
✞
1(%i5) cauchy_matrix([a,b])
✆
(%o5)



1
2a
1
b+a
1
b+a
1
2b




108 CHAPTER 10. MATRIX
10.2.8 Characteristics Polynomial (charpoly)
charpolyreturns the characteristic polynomial of the matrixAwith respect to
variablex. The characteristics polynomial of a matrixAis given by
|A−λI|
The Maxima syntax equivalent tocharpolyis
✞
1(%i5) determinant(A - diagmatrix(length(A), x))
✆
✞
1(%i5)matrix([1,2],[3,4])$
(%i6) charpoly(%,x);
✆
✞
(%o5) (1-x)(4-x)-6
✆
10.2.9 Coeﬃcient Matrix (coefmatrix)
coefmatrixreturns the coeﬃcient matrix for the variablesx1,x2,. . .,xmof the
system of linear equationseqn1,. . .,eqnm. In otherwords, each equation forms
one row of the matrix and its coeﬃcients construct columns for that row. The
syntax is
✞
1(%i6) coefmatrix(
<equaion lists>,
3 [<firstvariable>, <secondvariable>]
);
✆
✞
(%i5) A:[2*x + 3*y=4, -x + 6*y=10]$
2(%i6) coefmatrix(A, [x,y]);
✆
(%o6)

2 3
−1 6

It diﬀers to theaugcoefmatrixas it ignores the linear equation equality value.
10.2.10 Columns of Matrix (col)
colreturns thei
th
column from a matrixA. The value ofimust be less than
or equal to the size of column of the matrixA. The return value is a matrix.
The syntax for the functioncolis
✞
(%i1) col(<matrix>, <column number>);
✆

10.2. INVERSE OF MATRIX 109
10.2.11 Column Vectors (columnvector)
It returns the column vector of size provided as argument to the functioncolum-
nvector. The syntax used is
✞
1(%i1) columnvector(<number of rows>);
✆
Another functioncovectcan also be used for the same purpose.
10.2.12 Copy Matrix (copymatrix)
It returns a copy of the matrixA. This is the only way to make a copy aside
from copyingAelement by element. The syntax used is
✞
1(%i1) copymatrix(<matrixname>);
✆
10.2.13 Determinant Of Matrix (determinant)
determinantcomputes the determinant value of a matrixA. Symbolically it is
represented by|A|.
✞
1(%i1) A:matrix([2,1,1],[1,2,4],[1,3,4]);
(%i2) determinant(A);
✆
✞
(\%o2) -7
✆
10.2.14 Determinant As Fraction (detout)
By default the variabledetoutis ‘false’. Whendetoutis ‘true’, the determinant
of a matrix whose inverse is computed is factored out of the inverse. For this
switch to have an eﬀect ofdoallmxopsanddoscmxopsand these switches should
be set to ‘false’.
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2])$
(%i6) detout:true$
3(%i7) doallmxops:false$
(%i8) doscmxops:false$
5(%i6)invert(A);
✆
(%o7)


−6 8 1
2−13 10
11 6 −7


31

110 CHAPTER 10. MATRIX
10.2.15 Diagonal Matrix (diagmatrix)
It returns a diagonal matrix of sizen×nwith the diagonal elements all equal
tox. The syntax is
✞
1(%i1) diagmatrix(<squarematrixsize>, <diagonal variable x>)
✆
For example
✞
1(%i5) diagmatrix(3,"a")
✆
(%o5)


a0 0
0a0
0 0a


10.2.16 Do All Matrix Operations (doallmxops)
Whendoallmxopsis ‘true’, all operations relating to matrices are carried out.
When it is ‘false’ then the setting of the individualdotswitches govern which
operations are performed. By defaultdoallmxopsis ‘true’.
10.2.17 Do Exponent Operations (domxexpt)
Whendomxexptis ‘true’, a matrix exponential,exp(A)whereAis a matrix, is
interpreted as a matrix with element [i, j] equals toexp (M[i, j]). Otherwise
exp(A)evaluates toexp (ev(A)).domxexptaﬀects all expressions of the form
base
power
where ‘base’ is an expression assumed scalar or constant, and ‘power’
is a list or matrix.
10.2.18 Do Matrix-Matrix Operations (domxmxops)
Whendomxmxopsis ‘true’, all matrix-matrix or matrix-list operations are car-
ried out (but not scalar-matrix operations); if this switch is ‘false’ such opera-
tions are not carried out. By default it is “false’.
10.2.19 Do Matrix Products (domxnctimes)
Whendomxnctimesis ‘true’, non-commutative products of matrices are carried
out. By default it is “false’.
10.2.20 Do Not Factorize (dontfactor)
dontfactormay be set to a list of variables with respect to which factoring is
not to occur.

10.2. INVERSE OF MATRIX 111
10.2.21 Do Scalar Matrix Operations (doscmxops)
Whendoscmxopsis ‘true’, scalar-matrix operations are carried out. By default
it is ‘false’.
10.2.22 Do Scalar Matrix Operation as Matrix (doscmx-
plus)
Whendoscmxplusis ‘true’, scalar-matrix operations yield a matrix result. This
switch is not subsumed underdoallmxops. By default it is ‘false’.
10.2.23 Product of Zero & Non-Scalar Term (dot0nscsimp)
By defaultdot0nscsimpis ‘true’. Whendot0nscsimpis ‘true’, a non-commutative
product of zero and a nonscalar term is simpliﬁed to a commutative product.
10.2.24 Product of Zero & Scalar Term (dot0simp)
By default it is ‘true’. Whendot0simpis ‘true’, a non-commutative product of
zero and a scalar term is simpliﬁed to a commutative product.
10.2.25 Product of One & Other Term (dot1simp)
By default it is ‘true’. Whendot1simpis ‘true’, a non-commutative product of
one and another term is simpliﬁed to a commutative product.
10.2.26 Associative of Dot Product (dotassoc)
By default it is ‘true’. Whendotassocis ‘true’, an expression (A·B)·Csimpliﬁes
toA·(B·C).
10.2.27 Product of Constant & Other Term (dotconstrules)
By default it is ‘true’. Whendotconstrulesis ‘true’, a non-commutative product
of a constant and another term is simpliﬁed to a commutative product.
10.2.28 Distribution in Dot Product (dotdistrib)
By default it is ‘false’. Whendotdistribis ‘true’, an expressionA·(B+C)
simpliﬁes toA·B+A·C.
10.2.29 Simpliﬁed to Exponent (dotexptsimp)
By default it is ‘true’. Whendotexptsimpis ‘true’, an expressionA·Asimpliﬁes
toAˆˆ2.

112 CHAPTER 10. MATRIX
10.2.30 Zero Exponsnt (dotident)
Default value ofdotidentis ‘1’.dotidentis the value returned byXˆˆ0.
10.2.31 Dot Screw Rules (dotscrules)
By defaultdotscrulesis ‘false’. Whendotscrulesis true, an expressionA·Cor
C·Asimpliﬁes toC∗AandA·(C∗B) simpliﬁes toC∗(A·B).
10.3 Matrix Characteristics
10.3.1 Echelon Form of Matrix (echelon)
echeloncomputes and returns the echelon form of a matrix by using Gaussian
row elimination method. The echelon form is computed from matrixAby
elementary row operations such that the ﬁrst non-zero element ineach row in
the resulting matrix is one and the column elements under the ﬁrst one in each
row are all zero. The matrix used in this function is of the form ofm×n. The
syntax used is
✞
1(%i2) echelon(<matrixname>);
✆
For example
✞
1(%i1) A:matrix([2,0,0],[2,2,4],[2,2,4])$
(%i2) echelon(A);
✆
(%o2)


1 0 0
0 1 2
0 0 0


10.3.2 Eigenvalues (eivals)
The eigenvalue equation for a matrixAis
Av−λv= 0
which is equivalent to
(A−λI)v= 0
whereIis then×nidentity matrix. All possible values ofλare called eigenvalues
of the given matrix. The functioneivalsevaluates the eigenvalues of the given
matrix. This function returns two list of values. The ﬁrst sublist of the return
value is the list of eigenvalues of the matrix, and the second sublist is the list of
the multiplicities of the eigenvalues in the corresponding order.

10.3. MATRIX CHARACTERISTICS 113
✞
(%i1) A:matrix([2,0,0],[2,2,4],[2,2,4]);
2(%i2) eivals(A);
✆
✞
(%o2) [[0,2,6], [1,1,1]]
✆
10.3.3 Eigenvectors (eivects)
Computes eigenvectors of the matrixA. The return value is a list of two ele-
ments. The ﬁrst is a list of the eigenvalues ofAand a list of the multiplicities
of the eigenvalues. The second is a list of lists of eigenvectors. There is one list
of eigenvectors for each eigenvalue. There may be one or more eigenvectors in
each list.
✞
1(%i1) A:matrix([2,0,0],[2,2,4],[2,2,4])$
(%i2) eivals(A)$
3(%i3) eivects(A);
✆
✞
(\%o2) [[0,1,-1/2],[1,-1/2,-1/2],[0,1,1]]
✆
10.3.4 Except Matrix (ematrix)
The syntax used inematrixis
✞
1(%i1) ematrix(<rows>, <cols>, <value>, <rowpos>, <col pos>)
✆
It constructs a matrixAwhose all values are zero except thatAij=x. Where
x,iandjrepresents to the ‘value’, ‘row pos’ and ‘col pos’ respectively. For
example
✞
1(%i1) ematrix(3,3,a,2,2)
✆
(%o1)


0 0 0
0a0
0 0 0


10.3.5 User Matrix (entermatrix)
entermatrixreturns a matrix of sizem×nby accepting elements from the user
interactively. The syntax is
✞
1(%i1) entermatrix(<rows>, <cols>)
✆
After pressing enter key, total elements equals tom×nare asked to enter by
user in the console and laterly a matrix is returned using these entered elements.
This function is useful to create a matrix of own choice.

114 CHAPTER 10. MATRIX
10.3.6 Generate Matrix (genmatrix)
genmatrixgenerates a matrix of given size by using a function. The syntax of
functiongenmatrixis
✞
1(%i2)genmatrix(<gen function>, <rows>, <cols>)
✆
For example
✞
1(%i1) h[i,j]:=i+j$
(%i2)genmatrix(h,3,3)
✆
(%o2)


2 3 4
3 4 5
4 5 6


10.3.7 Identity Matrix (ident)
identstands for identity matrix. It generates a square matrix of sizenwhose
other than diagonal elements are zero. Syntax foridentis
✞
(%i1}ident(<squarematrixsize>)
✆
10.3.8 Inverse Matrix (invert)
invertreturns the inverse matrix of square matrixA. IfAis normal square
matrix then inverse matrix isA
−1
and relation between normal square matrix
and inverse matrix isA·A
−1
=I. A matrix is invertible if|A| 6= 0 or constant
terms in its characteristics polynomial is not zero.
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2])$
(%i6)invert(A)
✆
(%o6)







−
6
31
8
31
1
31
2
31
−
13
31
10
31
11
31
6
31
−
7
31








10.3. MATRIX CHARACTERISTICS 115
10.3.9 List Elements of Matrix (listmatrixentries)
It returns the elements of a matrixAin form of list.
✞
(%i1) A:matrix([2,0,0],[2,2,4],[2,2,4])$
2(%i2) list_matrix_entries(A);
✆
✞
(\%o2) [2, 0, 0, 2, 2, 4, 2, 2, 4]
✆
10.3.10 Left Side Delimiter (lmxchar)
This variable sets the left side delimiter of the matrix. Default value is ‘[’.New
left side matrix delimiter is set by using this variable like
✞
1(%i1)lmxchar:"|"% A vertical line
✆
10.3.11 Right Side Delimiter (rmxchar)
This variable sets the right side delimiter of the matrix. Default value is‘]’.
New right side matrix delimiter is set by using this variable like
✞
1(%i1)rmxchar:"|"% A vertical line
✆
10.3.12 Adjoin Matrix (adjoin)
It returns the union of the setAwith matrix or listB.adjoincomplains ifA
is not a literal set.adjoinandunionare equivalent; however,adjoinmay be
somewhat faster thanunion.
✞
1(%i5) adjoin (c, {a, b});
(%i6) adjoin (a, {a, b});
✆
✞
(\%o5) {a, b, c}
(\%o6) {a, b}
✆
10.3.13 Transpose of a Matrix (transpose)
Transpose of a matrix is the replacement of rows with columns or viceverse. It
is represented byA
′
. Determinant of matrixAand its transpose are equal.
✞
(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
2(%i6)transpose(A)
✆

116 CHAPTER 10. MATRIX
(%o6)


1 4 5
2 1 4
3 2 2


10.3.14 Mapping of Matrix (matrixmap)
10.3.15 Expression As Matrix (matrixp)
Returns ‘true’ if the expression is a matrix. Syntax for the functionmatrixpis
✞
(%i1)matrixp(<matrixname>)
✆
10.3.16 Trace of Matrix (mattrace)
mattracereturns the trace of the square matrixA. Tracing of a matrix is the
sum of all elements in main diagonal of the matrixA. To evaluate tracing of
matrix, ﬁrst load the “nchrpl” macro.
✞
1(%i4)load("nchrpl");
(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
3(%i6)mattrace(A);
✆
✞
(%o6) 4
✆
10.3.17 Minor of Matrix (minor)
minorreturns theMijminor of a matrixA. The syntax used is
✞
1(%i6)minor(<matrix>, <row>, <column>);
✆
MinorMij, of a matrixAis obtained by removingi
th
row andj
th
column. For
example
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
(%i6)minor(A,1,1)
✆
(%o6)

1 2
4 2

10.3. MATRIX CHARACTERISTICS 117
10.3.18 Determinant of Matrix (newdet)
newdetcomputes the determinant of the matrixAby the Johnson-Gentleman
tree minor algorithm.newdetreturns the result in CRE form.
10.3.19 Permanent of Matrix (permanent)
permanentcomputes the permanent of the matrixAby the Johnson-Gentleman
tree minor algorithm. A permanent is like a determinant but with no sign
changes.permanentreturns the result in CRE form.
10.3.20 Rank of Matrix (rank)
Rank of a matrix is the number of non zero rows of a matrix when it is simpliﬁed
in echelon form. Rank of matrix
A=

1 2
4 2

is ‘2’ while rank of matrix
A=


1 2 3
1 1 4
0 0 0


is also ‘2’ though matricesAandBare of diﬀerent orders. The syntax for
functionrankis
✞
(%i3)rank(<matrix>)
✆
✞
1(%i1) A:matrix([2,2,3],[2,4,3],[4,7,4]);
(%i2)rank(A);
✆
✞
(%o2) 3
✆
10.3.21 Rationalize to Matrix (ratmx)
Whenratmxis ‘false’, determinant and matrix addition, subtraction, and mul-
tiplication are performed in the representation of the matrix elements and cause
the result of matrix inversion to be left in general representation.
10.3.22 Row of Matrix (row)
rowreturns thej
th
row from a matrixA. The value ofjmust be less than or
equal to the size of row of the matrixA. The return value is a matrix. The
syntax for the functionrowis
✞
1(%i1)row(<matrix>, <rownumber>);
✆

118 CHAPTER 10. MATRIX
10.3.23 Solve To Scalar Matrix (scalarmatrixp)
By default it is ‘true’. Whenscalarmatrixpis ‘true’, then whenever a 1×1
matrix is produced as a result of computing thedotproduct of matrices it is
simpliﬁed to a scalar, namely the sole element of the matrix.
10.3.24 Sub Matrix (submatrix)
submatrixreturns the sub matrix from a given matrix by deleting rows and
columns supplied as arguments. Syntax for this function is
✞
1(%i1)submatrix(
<rownumber as integers>,
3 <matrix>,
<col number as integers>
5 );
✆
For example
✞
1(%i5) A:matrix([1,2,3],[4,1,2],[5,4,2]);
(%i6)submatrix(1,A,1)
✆
(%o6)

1 4
2 2

10.3.25 Triangularization of Matrix (triangularize)
triangularizereturns the upper triangular form of the matrixA, as produced
by Gaussian elimination. The return value is the same as echelon, except that
the leading nonzero coeﬃcient in each row is not normalized to 1. Syntax for
the function is
✞
(%i1)triangularize(<matrix>);
✆
10.3.26 Zero Matix (zeromatrix)
zeromatrixreturns a matrix of sizem×nwhose all elements are zeros. Syntax
for this function is
✞
1(%i1)zeromatrix(<rows>, <cols>)
✆
10.3.27 Matrix Element Transpose (matrixelementtranspose)
matrixelementtransposeis the operation applied to each element of a matrix
when it is transposed. Default value ofmatrixelementtransposeis ‘false’.

10.4. SOLUTION OF MATRIX 119
10.3.28 Matrix Element Multiplication (matrixelementmult)
matrixelementmultis the operation invoked in place of multiplication in a ma-
trix multiplication.matrixelementmultcan be assigned any binary operator.
Default value is “*”.
10.3.29 Matrix Element Addition (matrixelementadd)
matrixelementaddis the operation invoked in place of addition in a matrix
multiplication.matrixelementaddcan be assigned any n-ary operator. The
default value is “+”.
10.4 Solution of Matrix

120 CHAPTER 10. MATRIX

Chapter 11
Array
arrayfunction creates an array having user deﬁned elements. The syntax for
the functionarrayis
✞
1(%i1) array(<array name>, <dim 1>, ..., <dim n>)
✆
and it creates one array having ‘n’ elements. More than one arrayscan be
created ifarrayfunction is syntaxed as
✞
1(%i1) array([<arr name 1>, ..., <arr name n>], <dim 1>, ..., <dim n>)
✆
and it creates ‘n’ arrays having same elements.
121

122 CHAPTER 11. ARRAY

Chapter 12
Special Functions
Following functions are known as special functions.
12.1 airy
12.1.1 airyai
12.1.2 airybi
12.1.3 airydai
12.1.4 airydbi
12.2 Bessel Function
Bessel function are classiﬁed in four forms those are given below.
12.2.1 Bessel Function of First Kind (besselj)
Bessel functions are the canonical solutionsy(x) of Bessel’s diﬀerential equation
x
2
d
2
y
dx
2
+x
dy
dx
+ (x
2
−α
2
)y= 0
for an arbitrary complex numberα(the order of the Bessel function). The most
important cases forαare as integer or half-integer. Bessel functions of the ﬁrst
kind, denoted asJα(x), are solutions of Bessel’s diﬀerential equation that are
ﬁnite at the origin (x = 0) for integer or positiveα, and diverge asxapproaches
zero for negative non-integerα. It is possible to deﬁne the function by its Taylor
series expansion around x = 0
Jα(x) =
∞
X
m=0
(−1)
m
m! Γ(m+α+ 1)

x
2

2m+α
123

124 CHAPTER 12. SPECIAL FUNCTIONS
where Γ(z) is the gamma function. The syntax for Bessel function of ﬁrst kind
is
✞
1(%i5) ans=bessel_j (alpha, x)
✆
The example is
✞
1(%i5) bessel_j(1.5,2)
✆
✞
(%o5) 0.49129377868716
✆
12.2.2 Bessel Function of Second Kind (bessely)
The Bessel functions of the second kind, denoted byYα(x) are solutions of the
Bessel diﬀerential equation that have a singularity at the origin (x= 0). These
are sometimes called Weber functions after Heinrich Martin Weber, and also
Neumann functions after Carl Neumann.
Yα(x) =
Jα(x) cos(απ)−J−α(x)
sin(απ)
✞
1(%i5) ans=bessel_y (alpha, x)
✆
The example is
✞
1(%i5) bessel_y(1.5,2)
✆
✞
(%o5) -0.3956232813587
✆
12.2.3 Hyperbolic Bessel Function of First Kind (besseli)
This is modiﬁed form of Bessel function (hyperbolic Bessel functions) of ﬁrst
kind.
Iα(x) =i
−α
Jα(ix) =
∞
X
m=0
1
m!Γ(m+α+ 1)

x
2

2m+α
The syntax is
✞
1(%i5) ans=bessel_i (alpha, x)
✆
Example is
✞
1(%i5) bessel_i(1.5,2)
✆
✞
(%o5) 1.09947318863311
✆

12.3. BETA FUNCTION (BETA) 125
12.2.4 Hyperbolic Bessel Function of Second Kind (besselk)
This is modiﬁed form of Bessel function (hyperbolic Bessel functions) of second
kind.
Kα(x) =
π
2
I−α(x)−Iα(x)
sin(απ)
The syntax is
✞
1(%i5) ans=bessel_k (alpha, x)
✆
Example is
✞
1(%i5) bessel_k(1.5,2)
✆
✞
(%o5) 0.17990665795209
✆
12.2.5 Expansion of Bessel Function (besselexpand)
If variable ‘x’ is not a number and order is half of an odd integer
1
then evaluation
of bessel functions is not perfomed due tobesselexpandis ‘false’ by default. For
example
✞
1(%i5) bessel_k(3/2,x)
✆
✞
(%o5) bessel_k(3/2,x)
✆
Ifbesselexpandis set to ‘true’ then the bessel function is expanded in terms of
other elementary functions. For example
✞
1(%i4) besselexpand:true;
(%i5) bessel_k(3/2,x)
✆
(%o6)
√
π
−
1
x
+ 1
·
e
−x
√
2
√
x
Whenbesselexpandis set to ‘true’, Bessel function returns a new identity that
can be used to evaluate various results by putting the value of ‘x’.
12.3 Beta Function (beta)
A beta function is deﬁned as
B(a, b) =
ΓaΓb
Γ(a+b)
1
remember in rational p/q form

126 CHAPTER 12. SPECIAL FUNCTIONS
✞
(%i6) beta(1,2)
✆
(%o6)
1
2
12.4 Gamma Function (gamma)
A Gamma function is represented by
Γ(z) =
Z
∞
0
t
z−1
e
−t
dt
The syntax is
✞
1(%i6)gamma(<a numberorfraction>)
✆
The normalized gamma function
γ(x, a) =
1
Γ(a)
Z
x
0
t
a−1
e
−t
dt
is computed by using the functiongammaincin the syntax like
✞
1(%i6)gamma(0.25)
✆
✞
(%o6) 3.625609908221908
✆
12.5 Euler Function (euler)
Returns then
th
Euler number for nonnegative integer ‘n’. The syntax used is
✞
1(%i6) euler(<a non-negative number>)
✆
Example is
✞
1(%i6) euler(10)
✆
✞
(%o6) -50521
✆

12.6. ERROR FUNCTION (ERF) 127
12.6 Error Function (erf)
It computes the error function according to the relation
erf(z) =
2
√
π
Z
z
0
e
−t
2
dt
The syntax is
✞
1(%i6) erf(<a numberorfraction>)
✆
✞
1(%i6) erf(0.25)
✆
✞
(%o6) 0.27632639016824
✆
12.7 Laplace Transform (laplace)
Laplace transform of a functionf(x) is given by
L[f(x)] =
Z
∞
0
f(x)e
−sx
dx
Where ‘x’ is old variable and ‘s’ is new variable. Functionlaplacereturns the
laplace transformation of a function. The syntax used is
✞
1(%i6)laplace(<expressionorfunction>, <old variable>, <new variable>)
✆
Example is
✞
1(%i4)laplace(sin(t), t, s);
✆
(%o4)
1
s
2
+ 1
12.7.1 Inverse Laplace Transform (ilt)
Functioniltis abbreviation of “inverse Laplace Transformation”. It is use to
get the inverse of Laplace Transform function. The syntax is
✞
1(%i6)ilt(<expressionorfunction>, <new variable>, <old variable>)
✆
Example is
✞
1(%i4)ilt(1/(x^2+1), s, t);
✆

128 CHAPTER 12. SPECIAL FUNCTIONS
(%o4) sin(t)
12.8 Lagrange Polynomial (lagrange)
12.9 Laguerre Polynomial (laguerre)
A Laguerre equation, of type
x
d
2
y
dx
2
+ (1−x)
dy
dx
+n y= 0
is a second-order linear diﬀerential equation. This equation has nonsingular
solutions only ifnis a non-negative integer. The associated Laguerre polynomial
is
Ln(x) =
e
x
n!
d
n
dx
n
−
e
−x
x
n
·
=
(
d
dx
−1)
n
n!
x
n
Laguerre polynomials are represented asLn(x). Functionlaguerrereturns the
Laguerre polynomial. The syntax used is
✞
1(%i6)laguerre(<degree n>, <independent variable x>)
✆
Example is
✞
1(%i4)laguerre(2, x);
✆
(%o4)
x
2
2
−2x+ 1
12.10 Legendre Polynomial
A Legendre equation is given by
d
dx
≤
(1−x
2
)
dy
dx
λ
+n(n+ 1)y= 0 n >0,|x|<1
and associated polynomial is given by
y=APn(x) +BQn(x) |x|<1
WherePn(x) is given by
Pn(x) =
1
2
n
n!
d
n
dx
n
×
(x
2
−1)
n
∗

12.10. LEGENDRE POLYNOMIAL 129
and it is Legendre polynomial of ﬁrst kind.Qn(x) is given by
Qn(x) =
1
2
Pn(x) ln
≤
1 +x
1−x
λ
and it is Legendre polynomial of second kind. There are two kinds of Legendre
Polynomials.
12.10.1 Legendre Polynomial of First Kind(legendrep)
The ﬁrst kind of Legendre polynomial is obtained by using functionlegendrep
like
✞
1(%i6) legendre_p(<degree n>, <independent variable x>)
✆
Example is
✞
1(%i4) legendre_p(2, x);
✆
(%o4)
3x
2
−1
2
12.10.2 Legendre Polynomial of Second Kind(legendre q)
The second kind of Legendre polynomial is obtained by using functionlegendreq
like
✞
1(%i6) legendre_q(<degree n>, <independent variable x>)
✆
Example is
✞
1(%i4) legendre_q(2, x);
✆
(%o4)
−
3x
2
−1
·
log

−
x+1
x−1

−6x
4

130 CHAPTER 12. SPECIAL FUNCTIONS
12.11 Jacobians
12.11.1 jacobi
12.11.2 jacobicd
12.11.3 jacobicn
12.11.4 jacobics
12.11.5 jacobidc
12.11.6 jacobidn
12.11.7 jacobids
12.11.8 jacobinc
12.11.9 jacobind
12.11.10 jacobins
12.11.11 jacobip
12.11.12 jacobisc
12.11.13 jacobisd
12.11.14 jacobisn
12.11.15 inversejacobicd
12.11.16 inversejacobicn
12.11.17 inversejacobics
12.11.18 inversejacobidc
12.11.19 inversejacobidn
12.11.20 inversejacobids
12.11.21 inversejacobinc
12.11.22 inversejacobind
12.11.23 inversejacobins
12.11.24 inversejacobisc
12.11.25 inversejacobisd
12.11.26 inversejacobisn
12.12 fourier
12.12.1 fourint
12.12.2 fourintcos
12.12.3 fourintsin
12.12.4 foursimp
12.12.5 foursin
12.13 elliptic
12.13.1 elliptice

Chapter 13
Graphics
13.1 Contour Plot (contourplot)
It plots the contours (curves of equal value) of expression overthe surface region.
Two dimensional contour plot syntax is like
✞
1(%i6) contour_plot(<expression>, <range 1>, <range 2>, ...., [<options>])
✆
Remember that the options are given like
✞
1[<option keyword>, <option value 1>, <option value 2>]
✆
Plotting of circle is given by
✞
1(%i6) contour_plot(x^2 + y^2, [x, -4, 4], [y, -4, 4])
✆
To set oﬀ the legends from the plot, use option [legend, false] like
✞
1(%i6) contour_plot(x^2 + y^2, [x, -4, 4], [y, -4, 4],[legend, false])
✆
131

132 CHAPTER 13. GRAPHICS
−4
−3
−2
−1
0
1
2
3
4
−4−3−2−1 0 1 2 3 4
(a)
x
2
+y
2
16
09
04
−4
−3
−2
−1
0
1
2
3
4
−4−3−2−1 0 1 2 3 4
(b)
Figure 13.1: (a) Contour plot of functionf(x) =x
2
+y
2
with radii of 2, 3 and
4 moving from inner to outer. (b) Same contour plot of functionf(xwithout
legends.
13.2 Plotting 2D Plot (plot2d)
plot2ddisplays a plot of one or more expressions as a function of one variable
or parameter. There are three ways of using ofplot2dfunction.
✞
1(%i6)plot2d(
<expr>,
3 <range 1>,
[<options>]
5 ) % For single plot range
✆
or
✞
1(%i7)plot2d(
[<expr 1>, <expr 2>...],
3 [<options>]
) % For multiple plot without range range
✆
or
✞
(%i8)plot2d(
2 [<expr 1>, <expr 2>...],
<range 1>,
4 [<options>]
) % For multiple plot with range
✆
There are two methods of two dimensional plot. First is “discrete” and second
is “parametric”. “Discrete” method can be used in two ways like
✞
1(%i6)plot2d(<expr>, [discrete, [x1, y1], [x2, y2], ...], [<options>])
✆

13.3. THREE DIMENSIONAL PLOTTING (PLOT3D) 133
or
✞
1(%i7)plot2d(
<expr>,
3 [discrete, [[x1, x2, ....], [y1, y2, ...]]],
[<options>]
5 )
✆
First method uses coordinate style while second method uses ‘x’ & ‘y’coordi-
nates separately. “Parametric” method is used like
✞
1(%i6)plot2d(
[parametric,
3 <x expr>,
<y expr>,
5 <t range>,
[<options>]]
7 )
✆
Example is
✞
1(%i6)plot2d([sin(x)], [x,-5,5]);
✆
13.3 Three Dimensional Plotting (plot3d)
plot3dplots a three dimensional plots. The function for three dimensionalplot
is given by
z=f(x, y)
Here, third dimension ‘z’ is calculated by using two dimensional value ‘x,’and
‘y’. The syntax for theplot3dis
✞
1(%i6)plot3d(
[<expr 1>, <expr 2>, ... ],
3 <x range>,
<y range>,
5 [<options>]
)
✆
13.4 Finding Plot Options (plotoptions)
This variable shows the list of all options used in plotting. The description of
these options is given below.
1.gridrepresents to the mess lines along x-axis and y-axis mutually perpen-
dicular to each others. The syntax forgridis like
✞
[grid, <integer>, <integer>]
✆

134 CHAPTER 13. GRAPHICS
2.axesare set either ‘true’ or ‘false’. If axis keyword is set to ‘true’ then
axes are visible in the plot otherwise invisible in the plot. The syntax for
axesis like
✞
1 [axes, true]%forshowing axes in plot
[axes, false]%forhiding axes in plot
3 [axes, <sumbols>]%foruser defined symbolic axes names
✆
3.colorsets the color of plot lines and plot points. There are seven primary
colors “blue”, “red”, “green”, “magenta”, “black” and “cyan”,The syntax
used is
✞
1 [color, <color name>]
✆
4.pointtypesets the plotting points in various forms. The point types are
“bullet”, “circle”, “plus”, “times”, “asterisk”, “box”, “square”, “triangle”,
“delta”, “wedge”, “nabla”, “diamond” and “lozenge”. The syntaxused is
✞
1 [point_type, <type name>]
✆
5.stylesets the style of plotting. The styles are “points” and “lines”. The
syntax used is
✞
1 [style, <style name>]
✆
6. Iflegendsis set ‘true’ then plot legends are shown in the plot ﬁgure oth-
erwise legends are not shown in the plot ﬁgure. User can assign legends
name by assigning a string value. The syntax used is
✞
1 [legends, <legends name>]%user define name
[legends, true]%showlegends
3 [legends, false]%hide legends
✆
7.xlabelrepresents the label name along the x-axis. The syntax used is
✞
1 [xlabel, <label name>]%user define label name
✆
8.ylabelrepresents the label name along the y-axis. The syntax used is
✞
1 [ylabel, <label name>]%user define label name
✆
9.zlabelrepresents the label name along the z-axis. The syntax used is
✞
1 [zlabel, <label name>]%user define label name
✆
10. Ifboxis set ‘true’ then box is shown around the plot ﬁgure and ifboxis
set ‘false’ then box around the plot ﬁgure is invisible. The syntax is

13.5. SET PLOT OPTIONS (SET PLOTOPTION) 135
✞
1 [box, true]%showbox around plot figure
[box, false]%hide box around plot figure
✆
11.sphericaltoxyzconverts spherical coordinates into cartesian coordinates.
The syntax is
✞
[transform_xy, spherical_to_xyz]
✆
12. Ifcolorboxis set ‘true’ then color box is shown around the plot ﬁgure and
ifcolorboxis set ‘false’ then color box around the plot ﬁgure is invisible.
The syntax is
✞
1 [colorbox, true]%showcolor box around plot figure
[colorbox, false]%hide color box around plot figure
✆
13. Iflogxis present in the option then x-axis is converts into logarithm scale.
The syntax is
✞
[logx]%logarithmic scaled x-axis
✆
14.ntickscontrols the number of plot points in theplot2dorplot3dplotting
function. The syntax is
✞
1 [nticks, <integer>]%plot points
✆
15. Axes ranges are controlled by ‘x’, ‘y’ and ‘z’ keyword. The syntax is
✞
1 [x, <minimum value>, <maximum value>]%x range
[y, <minimum value>, <maximum value>]%y range
3 [z, <minimum value>, <maximum value>]%z range
✆
16.elevation&azimuthare used to control the viewing point to a three
dimensional plot. The syntax is
✞
1 [elevation, <integer value from 0 to 360>]%elevation
[azimuth, <integer value from 0 to 360>]%azimuth
✆
13.5 Set Plot Options (setplotoption)
This function is used to set the plot options. The syntax used is
✞
(%i11) set_plot_option([<option keyword>, <opt value 1>,<opt value 2>,
...])
✆

136 CHAPTER 13. GRAPHICS
13.6 Spherical To Cartesian (sphericaltoxyz)
This variable is used to transform Spherical coordinates into Cartesian coordi-
nates.

Chapter 14
Binary Operations
14.1 AND Operation (and)
andis ann−aryinﬁx operator; its operands are Boolean expressions, and its
result is a Boolean value. It returns ‘true’ if all are ‘true’ logically. The global
ﬂagprederroroperand cannot be determined to be ‘true’ or ‘false’.andprints
an error message whenprederroris ‘true’. Otherwise, operands which do not
evaluate to ‘true’ or ‘false’ are accepted, and the result is a boolean expression.
X Y Z
14.2 OR Operation (or)
oris ann−aryinﬁx operator; its operands are Boolean expressions, and its
result is a Boolean value. It returns ‘true’ if either of all is ‘true’ logically. The
global ﬂagprederroroperand cannot be determined to be ‘true’ or ‘false’.or
prints an error message whenprederroris ‘true’. Otherwise, operands which
do not evaluate to ‘true’ or ‘false’ are accepted, and the result is aBoolean
expression.
X Y Z
14.3 NOT Operation (not)
notnot is a preﬁx operator; its operand is a Boolean expression, and itsresult
is a Boolean value. It inverts the supplied argument. The global ﬂagprederror
operand cannot be determined to be ‘true’ or ‘false’.notprints an error message
whenprederroris ‘true’. Otherwise, operands which do not evaluate to ‘true’
or ‘false’ are accepted, and the result is a Boolean expression.
137

138 CHAPTER 14. BINARY OPERATIONS
X Y Z

Chapter 15
File Operations
Following functions are useful for ﬁle operations.
15.1 Reading
Now open the ﬁle ‘ﬁlename’ and see what is stored.
15.1.1 Append Console Transcript to File (appendﬁle)
appendﬁleappends the console transcript to a ﬁle. The syntax used is
✞
1(%i11) appendfile(<filename>)
✆
During the appending, previous contents of the ﬁle ‘ﬁlename’ are not erased.
appendﬁlemust be closed bycloseﬁleat the end of session.
15.1.2 Read File & Evaluate (batch)
This function reads ﬁle and evaluates all the expressions. Scope ofthe function is
limited to those ﬁles which contains standard Maxima expressions andfunctions
or ﬁles written by the Maxima functionwriteﬁle. Syntax of the function is
✞
1(%i11) batch(<filename>)
✆
15.1.3 Type of File (ﬁletype)
ﬁletypereturns a guess about the content of ‘ﬁlename’, based on the ‘ﬁlename’
extension. ‘ﬁlename’ need not refer to an actual ﬁle; no attempt ismade to
open the ﬁle and inspect the content. The syntax used is
✞
1(%i11) file_type(<filename>)
✆
139

140 CHAPTER 15. FILE OPERATIONS
15.1.4 Evaluation From File From Directory (load)
loadevaluates expressions in ‘ﬁlename’, thus bringing variables, functions, and
other objects into Maxima. The synax used is
✞
1(%i11)load(<filename>)
✆
15.1.5 Evaluation From File (loadﬁle)
loadﬁleevaluates Lisp expressions in ﬁlename.loadﬁledoes not invokeﬁlesearch,
as much of the path as needed to ﬁnd the ﬁle.loadﬁlecan process ﬁles created
bysave,translateﬁleorcompileﬁle. The syntax used is
✞
1(%i11)loadfile(<filename>)
✆
15.1.6 File Load Path Name (loadpathname)
This variable tells the path from where a ﬁle was loaded by using the function
load. If, previously no ﬁle was loaded then it returns ‘false’ value.
15.1.7 File Load Print Message (loadprint)
By default it is ‘true’.loadprinttells whether to print a message when a ﬁle is
loaded.
1. Whenloadprintis ‘true’, always print a message.
2. Whenloadprintis “loadﬁle”, print a message only if a ﬁle is loaded by the
functionloadﬁle.
3. Whenloadprintis “autoload”, print a message only if a ﬁle is automatically
loaded.
4. Whenloadprintis ‘false’, never print a message.
15.1.8 Show File Contents In Console (printﬁle)
printﬁlereads a ﬁle and puts all the contents of the ﬁle into Maxima console.
The syntax for the function is
✞
1(%i11) printfile(<filename>, all)
✆
15.1.9 Evaluates Expression From File (batchload)
batchloadreads Maxima expressions from ﬁlename and evaluates them, with-
out displaying the input or output expressions and without assigninglabels to
output expressions. The syntax used is
✞
1(%i11) batchload(<filename>)
✆

15.2. WRITING 141
15.2 Writing
15.2.1 Write Transcript in File (writeﬁle)
writeﬁlecreats and starts a hook to write a transcript of the Maxima session
to the ﬁle speciﬁed as ‘ﬁlename’. All interaction between the user andMaxima
is then recorded in this ﬁle, just as it appears on the console. Remember that
the contents written to ﬁle can not be loaded back to Maxima. Afterthe end of
interaction, the ﬁle must be closed by using functioncloseﬁle. The syntax for
starting of writting of a ﬁle is
✞
1(%i1)writefile(<filename>)
✆
After initiation of hook, do some thing in Maxima and then use the function
closeﬁlelike
✞
1(%i10) closefile(<filename>)
✆
15.2.2 Write Expressions to File (stringout)
stringoutwrites the expressions to the ﬁle in the same form as they are written
in console.
✞
1(%i1)stringout(<filename>,<options>)
✆
The given options are “input”, “functions” and “values”. If optionis “input”
then all the inputs in the current session are written to the ﬁle name. Similarly,
options “functions” is for functions of the current session and “values” is for all
values of the current session to write in the ﬁle. For example
✞
1(%i2)stringout(<filename>,input)
✆
Writes all the inputs of current session to the ﬁle name ‘ﬁlename’. Expressions
can be directly written to the ﬁle by using syntax
✞
1(%i3)stringout(<filename>, <expr 1>, <expr 2>, ....)
✆
15.2.3 ﬁleoutputappend
ﬁleoutputappendgoverns whether ﬁle output functions append or truncate
their output ﬁle. Whenﬁleoutputappendis ‘true’, such functions append to
their output ﬁle. Otherwise, the output ﬁle is truncated. By default it is ‘false’.
15.2.4 Write Output To a File (withstdout)
It evaluates expressions and writes output to a ﬁle or output stream. The
evaluated expressions are not written to the output. Output maybe generated
by functionprint,displayorgrind. In otherwords, expression may be arguments
ofprintfunction. The syntax used is

142 CHAPTER 15. FILE OPERATIONS
✞
1(%i11) with_stdout(<file name>, <expr_1>, <expr_2>, <expr_3>, ...)
✆
For Example
✞
1(%i11) with_stdout("c.txt",print(integrate(x,x)) )
✆
It will write the output of the integration of function ‘x’ with respect to ‘x’ in
the ﬁle “c.txt”. To see the output, useprintﬁlefunction like
✞
1(%i11) printfile("c.txt")
✆
There will the output of the integration of function ‘x’ with respectto ‘x’ in the
console.
15.2.5 Save A File (save)
savefunction is used to save contents into a ﬁle in the current directoryor path.
If ‘ﬁlename’ is absolute path then ﬁle is saved in the directory of givenpath. If
‘ﬁlename’ is relative path then ﬁle is saved in current path. There aretwo main
methods of usingsavefunction.
✞
1(%i1)save(<filename>, <values>, <functions>, <labels>, ...)
✆
If ‘values’, ‘functions’ etc are processing identiﬁcations like (%i1), (%o1) etc
then these speciﬁc input output values will be saved in the ﬁle. Ifsavefunction
is used like
✞
1(%i1)save(<filename>, all)
✆
Then whole session will be saved in the ﬁle. For example
✞
1(%i7)save("a.txt",%i5,%o2);
✆
and see the ﬁle ‘a.txt’ in current working directory. Similarly second method is
✞
1(%i8)save("a.txt", all);
✆
Contents written to a ﬁle bysavefunction can be reloaded by using function
loadﬁle.
15.2.6 Close Opened File (closeﬁle)
closeﬁlecloses the previously opened ﬁle by functionwriteﬁle. Syntax is
✞
1(%i10) closefile(<filename>)
✆

15.3. SEARCH 143
15.3 Search
15.3.1 ﬁlesearch
ﬁlesearchfunction searches the path or ﬁle name and returns the path or ﬁle
name if found. If ﬁle is not found then it returns ‘false’. The syntax is
✞
1(%i11) file_search(<filename>, <pathlist>)
✆
If argument ‘pathlist’ is omitted then ﬁle serach scope is for current directory and
if ‘pathlist’ argument is supplied then ﬁle search is restricted to that‘pathlist’
only. For example
✞
1(%i11) file_search("/")
✆
The output is
✞
(%o11) false
✆
15.3.2 ﬁlesearchdemo
This variable shows the path of ﬁles of demo directoies. The syntax is
✞
1(%i11) file_search_demo;
✆
15.3.3 ﬁlesearchmaxima
This variable shows the path of ﬁles where Maxima is installed. The syntax is
✞
1(%i11) file_search_maxima;
✆
The ﬁle names are expressed in form of $$${<ext 1>,<ext 2>,<ext 3>,...}.
15.3.4 ﬁlesearchusage
This variable shows the ﬁles of user’s document. The syntax is
✞
1(%i11) file_search_usage;
✆
The ﬁle names are expressed in form of $$${<ext 1>,<ext 2>,<ext 3>,...}.
15.3.5 ﬁlesearchlisp
This variable shows the ﬁles of lisp’s directories. The syntax is
✞
1(%i11) file_search_lisp;
✆
The ﬁle names are expressed in form of $$${<ext 1>,<ext 2>,<ext 3>,...}.

144 CHAPTER 15. FILE OPERATIONS
15.3.6 ﬁlesearchtests
This variable shows the ﬁles of tests directories. The syntax is
✞
1(%i11) file_search_tests;
✆
The ﬁle names are expressed in form of $$${<ext 1>,<ext 2>,<ext 3>,...}.
15.3.7 ﬁletypelisp
This variable returns the list of lisp’s extensions.
15.4 Path
15.4.1 ﬁletypemaxima
This variable returns the list of Maxima’s extensions.
15.4.2 pathnamedirectory
pathnamedirectoryreturns the directory path from the supplied path name.
The syntax is
✞
1(%i11) pathname_directory(<pathname>)
✆
For example
✞
1(%i7) pathname_directory("C:/server/a.txt")
✆
✞
(%o7) C:/server/
✆
15.4.3 pathnamename
pathnamenamereturns the name of ﬁle from the supplied path name. The
syntax is
✞
1(%i11) pathname_name(<pathname>)
✆
For example
✞
1(%i7) pathname_name("C:/server/a.txt")
✆
✞
(%o7) a
✆

15.4. PATH 145
15.4.4 pathnametype
pathnametypereturns the extension of ﬁle from the supplied path name. The
syntax is
✞
1(%i11) pathname_type(<pathname>)
✆
For example
✞
1(%i7) pathname_type("C:/server/a.txt")
✆
✞
(%o7) txt
✆

146 CHAPTER 15. FILE OPERATIONS

Chapter 16
Polynomial
An algebraic relation whose degree is more than two are known as polynomials.
For example
F(x) =ax
3
+bx
2
+cx+d
is a polynomial of third degree.
16.1 Roots
16.1.1 All Roots of Polynomial (allroots)
This command computes all roots, either complex or real, of a polynomial. The
precision value of roots depends on thefpprecvalue. For example
✞
1(%i7) f:x^2-2*x+2$
(%i8) allroots(f);
✆
✞
(%o8) [ x=1.0i + 1.0, x=1.0 - 1.0i ]
✆
16.1.2 All Big Float Roots of Polynomial (bfallroots)
This command computes all roots, either complex or real, of a polynomial. The
roots are big ﬂoats. For example
✞
1(%i7) f:x^2-13*x+2$
(%i8) bfallroots(f);
✆
✞
(%o8) [x=1.557112297752398b-1, x=1.284428877022476b1 ]
✆
147

148 CHAPTER 16. POLYNOMIAL
16.1.3 All Roots Within Limits (nroots)
It returns the all of real roots of the real univariate polynomial ranges between
self open interval of (min, max]. The syntax for this function is
✞
1(%i1)nroots(<function>, <minimumlimit>, <maximumlimit>)
✆
For example
✞
1(%i7) f:x^3-4*x+2$
(%i8)nroots(f);
✆
✞
(%o8) 2
✆
16.1.4 Nth Root of Base (nthroot)
Ifpis number whosen
th
root isqasq=
n
√
pthen
q
n
=P
In Maxima,nthrootis used like
✞
1(%i1)nthroot(<base>, <nth root>)
✆
For example
✞
1(%i8)nthroot(16,4);
✆
✞
(%o8) 2
✆
If there is no perfectnthrootof ‘base’ then there will be a warning message that
“base is not a false-th power”. To see this message, execute following example.
✞
1(%i9)nthroot(15,4);
✆
16.1.5 Real Roots (realroots)
It computes the roots of a uni-variable polynomial and returns onlyreal roots.
A tolerance value can be supplied as argument for rationalization of the real
root. Remember that the coeﬃecients of polynomial should be literal numbers
not symbol constants such as %pi. Syntax used for this function is
✞
1(%i1)realroots(<expression>, <tolerance>)
✆
✞
1(%i7) f:x^2-4*x+2$
(%i8)realroots(f);
✆

16.2. FACTOR 149
(%o8)
≤
x=
19655731
33554432
, x=
114561997
33554432
λ
16.2 Factor
16.2.1 Separation of Coeﬃcients (bothcoef)
It returns a list whose ﬁrst member is the coeﬃcient of x in expression and
whose second member is the remaining part of expression. Assume an expression
A∗x+B. If it is called in functionbothcoefas
✞
(%i1) bothcoef(A*x + B, x)
✆
then it will return a list of both coeﬃcients as
✞
(%o1) [A, B]
✆
16.2.2 Coeﬃcient of Term (coeﬀ)
The syntax forcoeﬀis
✞
1(%i1) coeff(<expression>, <variable of expression>, <degree of term>)
✆
The scope ofcoeﬀis list, vector and matrix. If degree of term is not supplied
as argument, then by default it is assumed as ‘1’.
✞
1(%i1) coeff(2*x^2+1, x);
coeff(2*x^2+1, x, 1);
3 coeff(2*x^2+1, x, 2);
✆
✞
(%o1) 0
(%o2) 0
(%o3) 2
✆
16.2.3 Divide Expression (divide)
This function dives ﬁrst expression by second expression assumingthird argu-
ment as a eliminating variable. The syntax for this function is
✞
1(%i1) divide(<expression 1>, <expression 2>, <main variable>)
✆
It returns a list of two values, ﬁrst is quotient and second is remainder.
✞
1(%i1) divide(x-y,x+y);
✆

150 CHAPTER 16. POLYNOMIAL
✞
(%o1) [-1, 2x]
✆
Similarly
✞
1(%i1) divide(x-y,x+y,x);
✆
✞
(%o1) [-1, 2y]
✆
16.2.4 Eliminate Variables (eliminate)
It eliminates one or more variables from two or more multi-variable algebraic
relations. This function consists two lists of expression and variables. Syntax is
✞
1(%i1) eliminate([<expr 1>, <expr 2>, ...],[<var 1>, <var 2>, ...])
✆
For example
✞
1(%i1) f1:x^2*y-3*z+y$
(%i2) f2:x-z*y+4*z$
3(%i3) eliminate([f1,f2], [y,z]);
✆
(%o3)
"√
4x
4
+ 3x
3
+ 8x
2
+ 3x+ 4 + 2x
2
+ 2
3
#
16.2.5 Factor of Expression (factor)
It returns the factor of a polynomial or a number. It is used like
✞
1(%i1)factor(<expression>, <minimum poly>)
✆
It factor expression over the ﬁeld of rationals with an element adjoined whose
minimum polynomial is ‘minimum poly’. It checks following ﬂags before factor-
ing.
1. Iffactorﬂagis ‘false’ suppresses the factoring of integer factors of rational
expressions.
2. Ifdontfactormay be set to a list of variables with respect to which fac-
toring is not to occur.
3. Ifsavefactorsis ‘true’ causes the factors of an expression which is a product
of factors to be saved by certain functions in order to speed up later
factorizations of expressions containing some of the same factors.
4. Ifberlefactis ‘false’ then the Kronecker factoring algorithm will be used
otherwise the Berlekamp algorithm, which is the default, will be used.

16.2. FACTOR 151
5. Ifintfaclimia ‘true’ Maxima will give up factorization of integers if no
factor is found after trial divisions and Pollard’s rho method. If setto
‘false’ complete factorization of the integer will be attempted.
16.2.6 Rearrange Factors (factorout)
It rearranges the expression in form of sum of terms for a given variable. The
syntax used is
✞
1(%i1)factorout(<expression>, <variable>)
✆
16.2.7 Rearrange Factors of Sum (factorsum)
It tries to group terms in factors of expression which are sums intogroups of
terms such that their sum is factorable.factorsumcan recover the result of
✞
1(%i1)expand((x + y)^2 + (z + w)^2)
✆
but it can’t recover
✞
1(%i1)expand((x + 1)^2 + (x + y)^2)
✆
because the terms have variables in common. Syntax for this function is
✞
1(%i1)factorsum(<expression>)
✆
16.2.8 Greatest Common Divisor (gcd)
It returns the greatest common divisor of two univariable polynomials. Syntax
for this function is
✞
1(%i1)gcd(<poly 1>, <poly 2>, <var>)
✆
Here ‘var’ is that variable for which greatest common divisor will be computed.
If ‘var’ is not supplied as argument to thegcdthen ﬁrst variable will be consider
as ‘var’. If there is greatest common divisor of two expression thenit is returned
other wise ‘1’ is returned for all ‘var’. For example
✞
1(%i1) f:x^2+y*x+c$
(%i2) g:x*y + x + c$
3(%i3)gcd(f,g);
✆
✞
(%o3) 1
✆
Again
✞
1(%i1) f:x^2 + 2*x + 1$
(%i2) g:x^3 + 1$
3(%i3)gcd(f,g);
✆

152 CHAPTER 16. POLYNOMIAL
✞
(%o3) x+1
✆
16.2.9 Greatest Common Divisor (gcdex)
It returns a list [a, b, u] whereuis the greatest common divisor of univariable
polynomialsfandg, anduis equal toa∗f+b∗g. The syntax is
✞
1(%i3)gcdex(<poly 1>, <poly 2>, <var>);
✆
An example is
✞
1(%i1) f:x^2 + 2*x + 1$
(%i2) g:x^3 + 1$
3(%i3)gcdex(f,g);
✆
✞
(%o3) [-y+2, 1, 3*y+3]
✆
If the two polynomials are integer values then functionigcdexis used.
16.2.10 Gaussian Factor (gfactor)
gfactorcomputes factors of an expression over the Gaussian integer.
✞
1(%i3)gfactor(<polynomial>);
✆
16.2.11 Highest Power (hipow)
It returns the highest exponent of variable ‘x’ of a univariable polynomial. This
function is used like
✞
1(%i1)hipow(<expression>, <variable>)
✆
16.2.12 Lowest Power (lopow)
It returns the lowest exponent of variable ‘x’ of a univariable polynomial. This
function is used like
✞
1(%i1)lopow(<expression>, <variable>)
✆
16.2.13 Return The Coeﬃcient (ratcoef)
ratcoefreturns the coeﬃcient ofxhaving powernwhen used like
✞
1(%i1)ratcoef(<expression>, <variable>, <power>)
✆

16.3. EXPANSION 153
16.2.14 Remainder (remainder)
It returns the remainder of the polynomial ‘poly 1’ when divided by polynomial
‘poly 2’. This function is used as
✞
1(%i1)remainder(<poly 1>, <poly 2>)
✆
16.3 Expansion
16.3.1 rat
16.3.2 Return Denominator (ratdenom)
ratdenomreturns the denominator of the expression.
16.3.3 Expand Ratio (ratdenomdivide)
Whenratdenomdivideis ‘true’,ratexpandexpands a ratio in which the numera-
tor is a sum into a sum of ratios, all having a common denominator. Otherwise,
ratexpandcollapses a sum of ratios into a single ratio, the numerator of which
is the sum of the numerators of each ratio.
16.3.4 Derivative of Rational Number (ratdiﬀ)
It diﬀerentiates the rational expression with respect tox. expression must be a
ratio of polynomials or a polynomial inx. The argumentxmay be a variable or a
subexpression of an expression. The result is equivalent todiﬀ, although perhaps
in a diﬀerent form.ratdiﬀmay be faster thandiﬀ, for rational expressions.
16.3.5 Rational Expansion (ratexpand)
It expands expression by multiplying out products of sums and exponentiated
sums, combining fractions over a common denominator, cancelling the greatest
common divisor of the numerator and denominator, then splitting the numerator
(if a sum) into its respective terms divided by the denominator.
16.3.6 Rational Factor (ratfac)
Whenratfacis ‘true’, canonical rational expressions (CRE) are manipulated in
a partially factored form.
16.3.7 Is Rational (ratp)
It returns ‘true’ if an expression is a rational expression.

154 CHAPTER 16. POLYNOMIAL
16.3.8 Rational Simpliﬁcation (ratsimp)
It simpliﬁes to the rational function. The result is returned as the quotient of
two polynomials in a recursive form, that is, the coeﬃcients of the main variable
are polynomials in the other variables.
16.3.9 Simplify Rational Exponent (ratsimpexpons)
When it is set to “true’ratsimpsimplify the rational function by using expo-
nents.
16.3.10 Rational Substitution (ratsubst)
ratsubstis used like
✞
1(%i1)ratsubst(a, b, c)
✆
ratsubstsubstitutes ‘a’ for ‘b’ in ‘c’ and returns the resulting expression. ‘b’
may be a sum, product, power, etc.
16.3.11 Resultant (resultant)
The functionresultantcomputes the resultant of the two polynomials ‘poly 1’
and ‘poly 2’, eliminating the variable ‘x’. This function is used as
✞
1(%i1)resultant(<poly 1>, <poly 2>, <variable>)
✆

Chapter 17
power series
In this section, Taylor series & Poisson series are explained.
17.1 Binomial Series
17.1.1 Deﬁne Taylor Function (deftaylor)
deftaylordeﬁnes a function as Taylor function. The syntax for the functionis
✞
1(%i1) deftaylor(
<f>,
3 <function expression>,
<subscript var>,
5 <init value>,
<final value>
7 )
✆
A Taylor function may be algebraic or trigonometric expression.
17.1.2 Max Taylor Order (maxtayorder)
Whenmaxtayorderis ‘true’, then during algebraic manipulation of Taylor series,
taylortries to retain as many terms as are known to be correct. By default it
is ‘true’.
17.1.3 Power Series (powerseries)
Iff(x) is given by a convergent power series in an open interval centeredatb
in the complex plane, it is said to be analytic in this interval. Thus forxin this
description,fis given by a convergent power series
f(x) =
∞
X
n=0
an(x−b)
n
155

156 CHAPTER 17. POWER SERIES
powerseriesconverts a Taylor function in a series notation.
✞
1(%i1)powerseries(<f>, <variable>, <pole point>)
✆
17.1.4 Taylor Series (taylor)
The Taylor series of a real or complex-valued functionf(x) that is inﬁnitely
diﬀerentiable at a real or complex numberais the power series
f(a) +
f
′
(a)
1!
(x−a) +
f
′′
(a)
2!
(x−a)
2
+
f
(3)
(a)
3!
(x−a)
3
+· · ·
which can be written in the more compact sigma notation as
∞
X
n=0
f
(n)
(a)
n!
(x−a)
n
wheren! denotes the factorial ofnandf
(n)
(a) denotes then
th
derivative off
evaluated at the pointa. The derivative offof the zero order isfitself and
(x−a)
0
and 0! are both1. Whena= 0, the series is also called a Maclaurin
series. The functiontaylorexpands the expression in a truncated Taylor or
Laurent series of the variable ‘x’ around the point ‘a’, containing terms through
(x−a)
n
. Its syntax is
✞
1(%i1)taylor(
<f>,
3 <variable>,
<pole point>,
5 <upto terms having power n>
)
✆
For example
✞
(%i2)taylor((sqrt(x)), x, a, 4)
✆
(%o2)/T/
√
a+
√
a(x−a)
2a
−
(x−a)
2
8
√
a a
+
(x−a)
3
16
√
a a
2
−
5 (x−a)
4
128
√
a a
3
+...
17.1.5 taylorlogexpand
It is ﬂag that controls the expansion of logarithmic function. If it is ‘true’ then
all logarithms are expanded fully so that zero-recognition problemsinvolving
logarithmic identities do not disturb the expansion process. If it is ‘false’ then
expansion of necessary logarithms take place to obtain a formal power series.
By default it is ‘true’.

17.1. BINOMIAL SERIES 157
17.1.6 taylorordercoeﬃcients
This ﬂag controls the ordering of coeﬃcients in a Taylor series. By default it is
‘true’.
17.1.7 taylorsimpliﬁer
taylorcalls this function to simplify the coeﬃcients of the power series expres-
sion. Syntax for this function is
✞
1(%i1) taylor_simplifier(<expression>)
✆
17.1.8 taylortruncatepolynomials
Whentaylortruncatepolynomialsis ‘true’, polynomials are truncated based
upon the input truncation levels. Otherwise, polynomials input for taylor are
considered to have inﬁnite precision. By default this ﬂag is set to ‘true’.
17.1.9 Depth of Taylor Terms (taylordepth)
If there are not nonzero terms, taylor doubles the degree of theexpansion ofg(x)
so long as the degree of the expansion is less than or equal ton×2
taylordepth
.
17.1.10 Info of Taylor Series (taylorinfo)
Returns information about the Taylor series expression.
17.1.11 Check Series as Taylor (taylorp)
Returns ‘true’ if expr is a Taylor series, and ‘false’ otherwise.
17.1.12 Print Power Series Message (verbose)
When verbose is ‘true’,powerseriesprints progress messages. By default it is
‘false’.
17.1.13 Integer to Poisson (intopois)
It converts an integer into poisson encoding. Syntax for it is
✞
1(%i1)verbose(<expression>)
✆

158 CHAPTER 17. POWER SERIES
17.1.14 outofpois
The syntax for this function is
✞
1(%i1)outofpois(<A>)
✆
ConvertsAfrom Poisson encoding to general representation. IfAis not in
Poisson form, outofpois carries out the conversion, i.e., the return value is
✞
1(%i1)outofpois(intopois($<$A$>$))
✆
This function is thus a canonical simpliﬁed for sums of powers of sine and cosine
terms of a particular type.
17.2 Poisson Series
17.2.1 Poisson Diﬀerentiation (poisdiﬀ)
The syntax for this function is
✞
1(%i1)poisdiff(<A>, )
✆
Diﬀerentiates ‘A’ with respect to ‘B’.
17.2.2 Poisson Exponent (poisexpt)
The syntax for this function is
✞
1(%i1)poisexpt(<A>, )
✆
Functionally identical tointopois(AˆB). ‘B’ must be a positive integer.
17.2.3 Poisson Integration (poisint)
The syntax for this function is
✞
1(%i1)poisint(<A>, )
✆
Integrates in a similarly restricted sense (to poisdiﬀ). Non-periodicterms in ‘B’
are dropped if ‘B’ is in the trig arguments.
17.2.4 Poisson Limit (poislim)
poislimdetermines the domain of the coeﬃcients in the arguments of the trig
functions.

17.2. POISSON SERIES 159
17.2.5 Poisson Plus (poisplus)
The syntax for this function is
✞
1(%i1)poisplus(<A>, )
✆
It is functionally identical tointopois(A + B).
17.2.6 Poisson Simpliﬁcation (poissimp)
The syntax for this function is
✞
1(%i1)poissimp(<A>)
✆
This function converts ‘A’ into a Poisson series for ‘A’ in general representation.
17.2.7 Poisson Substitution (poissubst)
The syntax for this function is
✞
1(%i1)poissubst(<A>, , <c>)
✆
This function substitutes ‘A’ for ‘B’ in ‘C’. ‘C’ is a Poisson series.
17.2.8 Poisson Multiplicaiton (poistimes)
The syntax for this function is
✞
1(%i1) poistimest(<A>, )
✆
It is functionally identical tointopois(A*B).
17.2.9 Print Poisson Series (printpois)
The syntax for this function is
✞
1(%i1)printpois()
✆
This function prints a Poisson series in a readable format. In commonwith
outofpois, it will convert a into a Poisson encoding ﬁrst, if necessary.

160 CHAPTER 17. POWER SERIES

Chapter 18
Sets
In this section, use of Maxima in Set is explained.
18.1 Properties of Sets
18.1.1 Distinct Elements of The Set (cardinality)
It returns the number of distinct elements of the set ‘A’. Syntax ofthe function
is
✞
1(%i1) cardinality(<set handle>)
✆
Example is
✞
1(%i4) A:{1,2,4,8,6,5,8,4,7}$
(%i5) cardinality(A);
✆
✞
(%o5) 7
✆
18.1.2 Disjoin of Sets (disjoin)
The syntax fordisjoinfunction is
✞
1(%i1) disjoin(<Set A>, <Set B>)
✆
disjoinreturns the set ‘B’ without the member of set ‘A’. If ‘A’ is not a member
of ‘B’, return ‘B’ unchanged. Example is
✞
1(%i4) A:{8,7}$
(%i5) B:{2,4}$
3(%i6) disjoin(A, B);
✆
✞
(%o6) {2,4}
✆
161

162 CHAPTER 18. SETS
18.1.3 Disjoint Sets (disjointp)
It returns ‘true’ if and only if both sets are disjoint.
18.1.4 Element of a Set (elementp)
It returns ‘true’ if and only if element ‘x’ is a member of set ‘A’ when this
function is used like
✞
1(%i1) elementp(<element>, <Set A>)
✆
Example is
✞
1(%i5) A:{2,4}$
(%i6) elementp(2, A);
✆
✞
(%o6) true
✆
18.1.5 Empty Set (emptyp)
It returns “true’ if and only if a set is an empty set.
✞
1(%i1) emptyp(<Set A>)
✆
18.1.6 External Subset (extremalsubset)
The syntax for this function is
✞
1(%i1) extremal_subset(<Set A>, <f>, <predicate>)
✆
It returns the subset of ‘A’ for which the function ‘f’ takes on predicate values.
The predicate value is either “minimum” or “maximum”.
18.1.7 ﬂatten
18.1.8 fulllistify
18.1.9 List From a Set (listify)
It returns a list containing the members of ‘A’ when ‘A’ is a set. Otherwise,
listifyreturns ‘A’. Syntax for this function is
✞
1(%i1)listify(<A>)
✆
Example is
✞
1(%i5) A:{2,4,6}$
(%i6)listify(A);
✆
✞
(%o6) [2,4,6]
✆

18.1. PROPERTIES OF SETS 163
18.1.10 Construct a Set (makeset)
✞
1(%i1)makeset(<expression>, <x>, <s>)
✆
Returns a set with members generated from the expression, where ‘x’ is a list
of variables in expression, and ‘s’ is a set or list of lists. To generate each set
member, expression is evaluated with the variables ‘x’ bound in parallel to a
member of ‘s’. For example
✞
1(%i1)makeset(j/k, [j,k], [[1,a],[3,v],[6,j],[a,b]])
✆
(%o1)

1
a
,
a
b
,
6
j
,
3
v
σ
Again
✞
1(%i1)makeset(k/j, [j,k], [[1,a],[3,v],[6,j],[a,b]])
✆
(%o1)

a,
b
a
,
j
6
,
v
3
σ
18.1.11 Partition of Set (partitionset)
partitionsetreturns a list of two sets. The ﬁrst set comprises the elements
of ‘A’ for which ‘predicate’ evaluates to ‘false’, and the second comprises any
other elements of set ‘A’.partitionsetdoes not apply to the return value of
‘predicate’. Syntax for this function is
✞
1(%i1) partition_set(<A>, <predicate>)
✆
18.1.12 Ordered Permutations of a Set (permutations)
This command returns a set of all distinct permutations of the members of the
list or set ‘A’. Each permutation is a list, not a set. When ‘A’ is a list, duplicate
members of ‘A’ are included in the permutations. The syntax of the function is
✞
1(%i1)permutations(<A>)
✆
18.1.13 Random Permutation (random permutation)
randompermutationreturns random permutations of the set or list ‘A’, as con-
structed by the Knuth shuﬄe algorithm.

164 CHAPTER 18. SETS
18.1.14 Check Set Equality (setequalp)
✞
1(%i1) setequalp(<A>, )
✆
returns ‘true’ if sets ‘A’ and ‘B’ have the same number of elements andis(x =
y)is true for ‘x’ in the elements of ‘A’ and ‘y’ in the elements of ‘B’.
18.1.15 Check For argument as Set (setp)
This function returns ‘true’ if argument is a Maxima set. Syntax forthis function
is
✞
1(%i1)setp(<Set A>)
✆
18.2 Operations On Sets
18.2.1 Maximum Element Value of a Set (lmax)
It returns the element wich has maximum value in a list or in a set. The syntax
is
✞
1(%i1) lmax(<A>)
✆
When ‘A’ is not a list or a set, Maxima shows error. Example is
✞
1(%i5) A:{2,4,6}$
(%i6) lmax(A);
✆
✞
(%o6) 6
✆
18.2.2 Minimum Element Value of a Set (lmin)
It returns the element wich has minimum value in a list or in a set. The syntax
is
✞
1(%i1) lmin(<A>)
✆
When ‘A’ is not a list or a set, Maxima shows error. Example is
✞
1(%i5) A:{2,4,6}$
(%i6) lmin(A);
✆
✞
(%o6) 2
✆

18.2. OPERATIONS ON SETS 165
18.2.3 Sub Set (subset)
The syntax for this function is
✞
1(%i1)subset(<Set A>, <predicate>)
✆
The argument ‘predicate’ controls the elemetns of subset being extracted from
the set ‘A’. Few examples of ‘predicate’ are “atom”, “evenp”, “oddp”, “min”,
“max” etc. For example
✞
1(%i1)subset({1,2,3,4,5,6}, evenp)
✆
✞
(%o1) [2, 4, 6]
✆
18.2.4 Predicate Subset (subsetp)
It returns ‘true’ if ‘A’ is a sub set of ‘B’ when this fucntion is used like
✞
1(%i1)subsetp(<A>, )
✆
18.2.5 Symmentric Diﬀerence (symmdiﬀerence)
It returns the symmetric diﬀerence of set ‘B’ from set ‘A’ if used like
✞
1(%i1)symmdifference(<A>, )
✆
18.2.6 Subtraction of Sets (setdiﬀerence)
Returns a set containing the elements in the set ‘A’ that are not in the set ‘B’.
Syntax for this function is
✞
1(%i1)setdifference(<A>, )
✆
18.2.7 Cartesian Product (cartesianproduct)
It returns the set of list of cartesian product of two sets. Syntax for this function
is
✞
1(%i1) cartesian_product(<Set A>, <Set B>)
✆
Example is
✞
1(%i4) A:{8,7}$
(%i5) B:{2,4}$
3(%i6) cartesian_product(A, B);
✆
✞
(%o6) {[7,2],[7,4],[8,2],[8,4]}
✆

166 CHAPTER 18. SETS
18.2.8 Intersection of Sets (intersection)
This function returns a set containing the elements that are common to the sets
‘A1’ through ‘An’.intersectioncomplains if any argument is not a literal set.
This function is used like
✞
1(%i1)intersection(<Set A>, <Set B>, ...., <Set Z>)
✆
Example is
✞
1(%i4) A:{2,4,6}$
(%i5) B:{3,4}$
3(%i6)intersection(A, B);
✆
✞
(%o6) {4}
✆
18.2.9 Power Set (powerset)
The function
✞
1(%i1)powerset(<A>)
✆
returns the set of all subsets of the set ‘A’. If cardinality of the set ‘A’ isnthen
powerset(<A>)has 2
n
members. The cardinality of the set is calculated by
using the functioncardinalityas
✞
1(%i1) cardinality(<A>)
✆
18.2.10 Union Of Sets (union)
It returns the union of two or more sets. This function is used like
✞
1(%i1)union(<Set A>, <Set B>, ...., <Set Z>)
✆
Example is
✞
1(%i4) A:{2,4,6}$
(%i5) B:{3,4}$
3(%i6)union(A, B);
✆
✞
(%o6) {2,3,4,6}
✆

Chapter 19
Statistics
Package “descriptive” contains a set of functions for making descriptive sta-
tistical computations and graphing. Following functions, commands, ﬂags and
variables are deﬁned in this package and to use it, ﬁrst load the “descriptive”
packege by using the syntax of functionloadlike
✞
1(%i1)load(descriptive)$
✆
19.1 Plotting
19.1.1 Bar Plot of Data (barsplot)
It plots bars diagrams for discrete statistical variables, both forone or multiple
samples. Syntax for this function is
✞
1(%i1) barsplot(<data 1>, ..., <data n>, <option 1>, ...., <option n>);
✆
Options, those can be used in bar plots are
1.boxwidth: width of bar rectangles.
2.grouping: indicates how multiple samples are shown. Valid values are:
“clustered” and “stacked”.
3.groupsgap: a positive integer number representing the gap between two
consecutive groups of bars.
4.barscolors: a list of colors for multiple samples. When there are more
samples than speciﬁed colors, the extra necesary colors are chosen at ran-
dom.
5.frequency: indicates the scale of the ordinates. Possible values are: “ab-
solute”, “relative”, and “percent”.
167

168 CHAPTER 19. STATISTICS
6.ordering: possible values are “orderlessp” or “ordergreatp”, indicating
how statistical outcomes should be ordered on the x-axis.
7.samplekeys: a list with the strings to be used in the legend. When the
list length is other than 0 or the number of samples, an error message is
returned.
8.startat: indicates where the plot begins to be plotted on the x axis.
All global draw options, exceptxtics, which is internally assigned by barsplot.
If, there is requirement for custom values for this option or build a complex
scenes, usebarsplotdescriptionas in following example
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) <list 2>:makelist(<rules of list>)$
(%i4) <bp descr 1>:barsplot_description(<list 1>, <optionvalues>)$
5(%i5) <bp descr 2>:barsplot_description(<list 2>, <optionvalues>)$
(%i6) draw(gr2d(<bp descr 1>), gr2d(<bp descr 2>))$
✆
19.1.2 Error Box Plot of Data (boxplot)
19.1.3 Stem Plot (stemplot)
19.1.4 Pie Chart (piechart)
19.1.5 Scatterin Plot of Data (scatterplot)
19.1.6 Asterisk Plot of Data (starplot)
19.1.7 Histogram Plot (histogram)
19.2 Regression & Correlation
19.2.1 Building a Sample (buildsample)
19.2.2 Correlation (cor)
19.2.3 Covariance Matrix (cov)
The covariance matrix of the multivariate sample, deﬁned as
S=
1
n
n
X
j=1
(Xj−
ˉ
X)(Xj−
ˉ
X)
′
WhereXjis the element of thej
th
row of a matrix. Syntax for this command
is

19.3. MEAN, MEDIAN & DEVIATION 169
✞
(%i1)load(descriptive)$
2(%i2) m:<amatrix>$
(%i3) cov(m)
✆
For example
✞
1(%i4) s: matrix([1,2,4],[8,6,5],[8,4,7])$
(%i5) cov(s);
✆
(%o4)










98
9
14
3
28
9
14
3
8
3
2
3
28
9
2
3
14
9










19.2.4 Covariance of Type One (cov1)
The covariance matrix of the multivariate sample, deﬁned as
S1=
1
n−1
n
X
j=1
(Xj−
ˉ
X)(Xj−
ˉ
X)
′
WhereXjis the element of thej
th
row of a matrix.
19.2.5 Coeﬃcient Variation (cv)
The variation coeﬃcient is the quotient between the sample standard deviation
(std) and the mean. It is obtained by functioncv. Syntax for this function is
✞
(%i4) cv(<listor matrix>)
✆
19.2.6 List Correlation (listcorrelations)
19.3 Mean, Median & Deviation
19.3.1 Average or Mean (mean)
The mean of samples of data is given by
ˉx=
1
n
n
X
i=1
xi
Syntax formeanis

170 CHAPTER 19. STATISTICS
✞
1(%i1)load(descriptive)$
(%i2) s:makelist(<rules of list>)$
3(%i3) mean(s),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) mean(s),numer;
✆
✞
(%o4) 5
✆
The samples may be in a list or in a matrix form.
19.3.2 Mean Deviation (mean deviation)
The mean deviation of samples of data is given by
dm=
1
n
n
X
i=1
|xi−ˉx|
Syntax formeandeviationis
✞
1(%i1)load(descriptive)$
(%i2) s:makelist(<rules of list>)$
3(%i3) mean_deviation(s),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) mean_deviation(s),numer;
✆
✞
(%o4) 2
✆
The samples may be in a list or in a matrix form.
19.3.3 Median (median)
Once the sample is ordered, if the sample size is odd the median is the central
value, otherwise it is the mean of the two central values.
✞
1(%i1)load(descriptive)$
(%i2) s:makelist(<rules of list>)$
3(%i3) median(s),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) median(s),numer;
✆
✞
(%o4) 5
✆

19.3. MEAN, MEDIAN & DEVIATION 171
19.3.4 Deviation from Median (median deviation)
The median deviation of samples of data is given by
Dm=
1
n
n
X
i=1
|xi−med|
Syntax formeandeviationis
✞
1(%i1)load(descriptive)$
(%i2) s:makelist(<rules of list>)$
3(%i3) median_deviation(s),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) median_deviation(s),numer;
✆
✞
(%o4) 2
✆
The samples may be in a list or in a matrix form.
19.3.5 Continuous Frequency (continuousfreq)
19.3.6 Discrete Frequency (discretefreq)
19.3.7 Geometric Means (geometric mean)
Geometric mean of given sample statistical data of lengthnis given by
GM=

n
Y
i=1
xi
!
1/n
The syntax used for this commmand is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) geometric_mean(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) geometric_mean(s),numer;
✆
✞
(%o4) 4.226198741306655
✆
The samples may be in a list or in a matrix form.

172 CHAPTER 19. STATISTICS
19.3.8 Harmonic Mean (harmonic mean)
Harmonic mean of given sample statistical data of lengthnis given by
HM=
n
P
n
i=1
1
xi
The syntax used for this commmand is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) harmonic_mean(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) harmonic_mean(s),numer;
✆
✞
(%o4) 3.261432269197584
✆
The samples may be in a list or in a matrix form.
19.3.9 Range of Data (range)
The range is the diﬀerence between the extreme values of a list or a matrix.
Syntax forrangeis
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) range(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) range(s),numer;
✆
✞
(%o4) 6
✆
19.3.10 Maximum Value of Sample List (smax)
It returns the maximum value of a sample list or matrix. When the argument
is a matrix,smaxreturns a list containing the maximum values of the columns,
which are associated to statistical variables. Syntax for thesmaxis
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) smax(<list 1>),numer;
✆

19.3. MEAN, MEDIAN & DEVIATION 173
19.3.11 Minimum Value of Sample List (smin)
It returns the minimum value of a sample list or matrix. When the argument
is a matrix,sminreturns a list containing the minimum values of the columns,
which are associated to statistical variables. Syntax for thesmaxis
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) smin(<list 1>),numer;
✆
19.3.12 Standard Deviation (std)
This is the the square root of variance when variance is calculated with denom-
inatorn.
SD=
√
σn
19.3.13 Standard Deviation Type One (std1)
This is the the square root of variance when variance is calculated with denom-
inatorn−1.
SD=
√
σn−1
19.3.14 Transforming Samples (tranform sample)
The syntax for this function is
✞
1(%i1)load(descriptive)$
(%i2) tranform_sample (<matrix>, <varlist>, <exprlist>)
✆
It transforms the sample matrix, where each column is called according to
‘varlist’, following expressions in ‘exprlist’.
19.3.15 Variance (var)
This is sample variance, deﬁned as
σ=
1
n
n
X
i=1
(xi−ˉx)
2
Syntax for thevaris
✞
(%i1)load(descriptive)$
2(%i2) <list 1>:makelist(<rules of list>)$
(%i3) var(<list 1>),numer;
✆
The sample may be a list or a matrix. For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) var(s),numer;
✆

174 CHAPTER 19. STATISTICS
✞
(%o4) 5.555555555555555
✆
19.3.16 Variance of Type One (var1)
This is sample variance, deﬁned as
σ=
1
n−1
n
X
i=1
(xi−ˉx)
2
Syntax for thevaris
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) var1(<list 1>),numer;
✆
The sample may be a list or a matrix. For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) var1(s),numer;
✆
✞
(%o4) 6.25
✆
19.4 Skew, Kurosis & Moment
19.4.1 Skewness of Data (skewness)
The coeﬀcient of skewness of given sample statistical data of lengthnis given
by
ks=
1
n s
3
n
X
i=1
(xi−ˉx)
3
The syntax used for this commmand is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) skewness(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) skewness(s),numer;
✆
✞
(%o4) -0.25455844122716
✆
The samples may be in a list or in a matrix form.

19.4. SKEW, KUROSIS & MOMENT 175
19.4.2 Pearson Skewness (pearsonskewness)
Coeﬃcient of Pearson’s Skewness is given by
Ps=
3(ˉx−med)
s
Heremedis median of the list. The sample data here may be in form of a
matrix. Syntax for this function is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) pearson_skewness(<list 1>),numer;
✆
For example
✞
1(%i4) s:[2,2,4,8,6,5,8,4,7]$
(%i5) pearson_skewness(s),numer;
✆
✞
(%o4) 0.14396315014974
✆
19.4.3 Quartile Skewness (quartileskewness)
The coeﬀcient of quartile skewness of given sample statistical dataof lengthn
is given by
Qs=
C
3/4−2C
1/2+C
1/4
C
3/4−C
1/4
WhereCpis thep−quantile of sample list. The syntax used for this commmand
is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) quartile_skewness(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) quartile_skewness(s),numer;
✆
✞
(%o4) 0.33333333333333
✆
The samples may be in a list or in a matrix form.
19.4.4 Kurtosis (kurtosis)
The coeﬀcient of kurtosis of given sample statistical data of lengthnis given by
k=
1
n s
4
n
X
i=1
(xi−ˉx)
4
−3
The syntax used for this commmand is

176 CHAPTER 19. STATISTICS
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) kurtosis(<list 1>),numer;
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) kurtosis(s),numer;
✆
✞
(%o4) -1.1352
✆
The samples may be in a list or in a matrix form.
19.4.5 noncentralmoment
19.4.6 Central Moment of Data (centralmoment)
Central moment of orderkof given statistical data about mean or average value
is given by
µ=
1
n
n
X
i=1
(xi−ˉx)
k
The syntax used for this commmand is
✞
1(%i1)load(descriptive)$
(%i2) <list 1>:makelist(<rules of list>)$
3(%i3) central_moment(<list 1>, <order k>)
✆
For example
✞
1(%i4) s:[1,2,4,8,6,5,8,4,7]$
(%i5) central_moment(s, 2);
✆
✞
(%o4) 50/9
✆
The samples may be in a list or in a matrix form.

Maxima CAS an introduction

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