HANDBOOK of MATHEMATICAL FORMULAs (compilation of equations).pdf

Handbook of math formulas; compilations of mathematical formulas; differentiation formulas; integration formulas; calculus; vectors; limits; fourier transforms; numerical analysis; statistics; random errors; series; matrix algebra; complex variables; differential equations; physical constants; hyperbolic functions

2. MATHEMATICAL FORMULA HANDBOOK

3. Contents

Introduction
Series
Arähmei and Game progresos Cameron sis, hero a

Tos: Binomial eons Ty

Vector Algebra

Matrix Algebra

Vector Calculus
Noto; Moni; Gi

Complex Variables

nr; De Molo’ then: Pour o

Trigonometric Formulae

Hyperbolic Functions

Limits
Differentiation,

Integration
Stn forms Stor atico; ne
Dine 6 ction Reduction form

Differential Equations
Laplace’ oti

Calculus of Variations
Functions of Several Variables
yor eis ort eras; Samy pons; Changing caries: th chain rae
Changing arabe in surja and ou negras bla

Fourier Series and Transforms
rier ers: Fourier seis fort

Faure sees or and wen nti
ke ers Fore sors

Laplace Transforms.

Numerical Analysis

ing 10 ders; relation: Ener om,

17. Treatment of Random Errors

Statistics
Mean and Vaio Pray distritos Weight
Saisies of date sample Regression (les

10

n

2

13
1

16

7
18

19

»

a

4. Introduction

Bibliography

brio Nand. aja

Abramowitz, Me Stegun, LA Handbook of Mathematical Functions, Dover, 1965.

GGradshteyn LS. & Ryzhik LM, Table o tra, Series and Products, Academic Pres, 1980

Jahnke, Ee Ede,

Norling Ce Osterman, J, Physics Handbook, Chartwell Brat, Bromley, 1960.

Speigel, MR, Mathematical Handy

(Schaum Outline Sri, MeGr Hi, 1968),

Physical Constants

Based om the “Review of Particle Pr

speed of light in a vacuum
permeability of a vacuum
permittivity of a vacuum
elementary charge

Planck constant

h/2n

Avogadro constant

unified atomic mass constant
mass of electron

mass of proton

Bohr magneton eh/Arım.
molar gas constant
Boltzmann constant
Stefan-Boltzmann constant
‘gravitational constant

Other dat

acceleration offre fll

arnet tl, 196, Physics Revie
Physical Constants, Cohen & Taylor, 1997, Physic Today, B
‘deviation uncertainties nthe ast ig)

2.997 924 58 x 10% ms!

(by definition)
4xx 10-7 Hm! (by definition)
8.854 187 817... x 10
19) x 107% €

6-626 075 5(40) x 107%] 5
¡00% y

"rm!

1.054 572 66(63) x
6.022 136 7(36) x 10
1.660 540 2(10) x 10°
9-109 389 7(54) x 10
1010) x 10°
9.274 015 4(31) x 10
314 510(70) IK! mol"

1-380 658(12) x 10-23 JK!

5.670 51(19) x 1078 wm? Kt
6.672 59(85) x 10-11 Nm? kg

9.806 65 ms"? (standard value at sea level)

5. 1. Series

Arithmetic and Geometric progress

(These results ala old for comple series)
Convergence of series: the ratio test

Convergence of series: the comparison test

each term in a series of positive terms i ess than the corresponding term in series Aou to

then the given seres als convergent
Binomial expansion

If is a postive integer the seres termina

es and valid for all the tem in x ") where °c

a the number of iffrent ways in which an unordered sample ofr objet on be selected from a set of

Taylor and Maclaurin Si

“act Tant Har

valid for a values of

valid oral values

valid for? < x < À

6. Integer series

En 142494046

Sram. m Mann

[sce expansion often! a]

Enon 2344 mena = MU 2
This ast result sa special case of the more general formula,
En Dr +2) = MACEDO 2) N AN +4 D)
Plane wave expansion
espía) = explitrcoso) = Fiat + tt Peux 0)
she (00) ar Legendre polynomial (os section 11) and (kr) ae spherical Bese functions, define by
2. Vector Algebra
IH, J Kane orthonormal veto and A = Ad + Ayj + A then JA]? = AE + A} + AR [Orthonarmal vectors

Scalar product

AB Alice wher isthe angle between ie vectors

‘Scalar multiplications commutative: AB = BA.
Equation of a line

A point = (x2) les ona ine pasting through point and parle

with Aa rea umber

7. Equation of a plane

{a)¢-d = dl here dis te normal from the origin othe plane, or

Vector product

Ax B = AI sing, wheredis the angle between th vectors and
Torwhich A, By form righthand st

nit vector normal tothe plane containing

A and Bin the dirt

Vestormoliplicaton not commutative: À X B = Bx A
Scalar triple product
Ac Ay A

Vector triple product

Non-orthogonal basis

Summation convention

8. 3. Matrix Algebra

Unit matrices

zen ix of order men Al = LA = A. Also =1

yj = bj A isa square

Products

cab) = Ea,

Transpose matrices

LEA is mati, then transpose matrix A such hat (A) = (A

Inverse matrices

transpose of cofactor of A

where the cofactor ofA s (1) times the determinant of the matrix A wid the th ow and ith column deleted

Determinants

Al= Ep AAA
‘where the numberof the sue sequal to the order of the matrix
2x2 matrices

was (2 8) sen

Product rules
AB. NINA (individual veses exis)
AB... NI = [Ai (individual matrices are square)

Orthogonal matrices

form

An orthogonal matrix Qs square matix whose colon
maté O

aan ja OF isalso

9. Solving sets of linear simultaneous equations

AE A is aquar then Ax = has unique solution x= AID A! exist if 1A] #0.
IFA is square then Ax = Ohas a non trivial solution and ony JA =,

Au overconsrsined et of equations Ax = bis one in which A has rows and cola, where {he number
‘of equations) i greater than (ih number of variable. The bes ac x (in Ihe sense hat minimizs the

cor (Ax — ] ithe solution ofthe equations AT Ax = ATE, Ifthe columns ofA are orthonormal vectors then

din components of 4.14 = AT then A is called a

conjugate ofthe corespo tan matrix

Eigenvalues and eigenvectors

The n eigenvalues A and eigenvectors, of an n xn mates Aare the solutions ofthe equation An = Au. T
‘genvales ae the zeros of the polynomial of gree, (A) = [A A IA is Herman then the elgevais
Ware eal and the eigenvectors ae mutually othogonal. [A — A} = Di all Ue characters equation of the

sholAI = TTA

11 Sis symmetric matrix, isthe diagonal matrix whose diagonal elements ar the eigenvalues ofS, and is the
eis whose caus ac the normalized eigenvectors of A then

If xis an approximation o an eigenvector a À then «TA (x) (Rayleigh q

int is an approximation to the
comesponding

Commutators.

Hermitian algebra

| Mat form Operator form Brake
Hermitity wre=tane fuos= fioure 1018)
Oxthogonalty OPEN (DEC

Completeness b= Env) °

Rayleigh-Ritz

Lowest eigenvalue x

10. Pauli spin matrices

ti dl BS
4, Vector Calculus

¿is ascalar function fast of positon cordinate, In Cartesian condinates
$= Sy.) in cylindrical polar coordinates $ = 6(9,9.2) in spheral
Pola coordinates $ = (70.9); cases with radial symmety 6

Mis veto function whose components are salar functions ol he positon
ar independent funcions of 2

In cartesian coordinates (et) al
endé- VO dwA=V-A culA=VxA

Identities
god +) = grado: + prod Au +A) = WA à di A
srad(dugh) = d1 grados + érgradé
cuna, + Aa) = cue ay + eu A
(HA) = Galva +(gradó)-A, eurl(pA) = peur + (grado)
divlaı x Ar) = Ar curl Ay = A curl
ur(As x As) = Audiv Az = Ardiv A + (ds grad) Ay — (Ay grad)
divicurlA) =0, — curigradg) =0
cul(cue A) = grad A) — div{grad A) = grad(aiv 4) - A
rad Ai» Aa) = Au x (cue As) + (As grad) As + A2 x (Cu A) + (Ar

11. Grad, Div, Curl and the Laplacian

Gradient vo

Transformation of integrals

L= the distance along some curve’ in space and is measured fom some fixed point
a volume specifi surfce
7 = the unit tangent to Ca the point P

AL = the vector element of curve (= FL)

INES

When 5 defines a lose region having a volume +

AS
[cora fous fivxae

12. osx ay-as= [aa

ax v9) as [oan

[ove-as= [v(wve

[were + (wo) «ve)) ar

[ore-or

#9) as

5. Complex Variables

Complex numbers

The complex number 2 = x
real quantity rs the modulus

De Moivre’s theorem

Power series for complex variables

This ast eres converges both on and within he ice

This ast series converges both on and within he ce

angle the argumento

1 exept atthe pint:

ceaceptat the points

cept at the point

The complex conjugate of zis "= ti

convergent for a rite

1

13. 6. Trigonometric Formulae

Arial Au At
sin(A£D) = sinAcosB ¿cosAsinE cosa

cos(A 28) = cos AmsBrsindsind — sinAsind

Relat

‚ns between sides and angles of any plane triangle

scribed cre

sind JO. wheres

Relations between sides and angles of any spherical triangle

14. 7. Hyperbolic Functions

= Forlage negative
Relations of the functions

sinh = —sioh(-a sechx sena
oh cost cosecha = —cosech(—»

hy o Ztani(a/2) tam ooh = Lan 6/2

Tan O72)” Yat ium?) ere

tanh = 4/1 seen sechx = tax

12/2) I osh(x/2) = ¡SER

5 tanh(2e) = Hats

22) = costa 114 2sinhty

15. 8. Limits

16. 9. Differentiation

10. Integration

Standard forms

fra teen #21
Jia mese finxaz =xins
ax = le [retó er (Eh) +e

sinrar= (in 3) +

/
/
/
/

17. Zand even

Land
fumar =o
feras =smx+e
“franhx dx = t(coshs
come x= In tan
fueras =Ingecr+uns) +¢ fans =2un He
foods nz fear = int) +6

fe

Standard substitutions

Ite integrand is function of sub

18. Integration by parts

f'udo= ul frau

Differenti

n of an integral

1 (tian abiteary function en 509/00 = fi

out 40,010 [7 sa

Reduction formulae

na coton = MEL [sino cont odd = MEL [Vino cor 2040

core be reduced venta

Y waa four [a=

19. 11. Differential Equations

Diffusion (conduction) equation

ERA
Wave equation

ve

Legendre's equation

mal), where Pi

solutions of which are Legendre poly

[ron ax

Bessel'sequation

Sy, dy
wat

20. Laplace’s equation

Ken

nd mis einer
1 expresedin hre-dimensionul pola coordinates (ee section 42 solution is

Where adm are intgers with > Im| > 0, B,C, Dar constants
cos) = sino [E] rican

trical polar cooninates (se section) sation

Spherical harmonics

Then

zed solutions 0,0)
2),

the quato

wenn en sens E eee
un Ev yr Eno eto

12. Calculus of Variations

5 al

The condition for 1 = | F(y. y) dto have a stationary values

Euler Lagrange equation.

rere y

21. 13. Functions of Several Variables

140 fi.) thon 2 np frein wih rpet ing costat
vom Mars Bays Mast. and so Bors Moy à
wher a independent variables. 22 o weten at (22) or 2] when the vais kept

constant need to be stated expicily

lisa ell behaved function then

19 = soy

a (2) (à) (à)

()

Taylor series for two variables

LE) well behaved inthe vi

sit) evn (ne)
wheres = a+ ny = 04 vand diferenciarlo, y =8

Stationary points

anti 6 = f(y) story pont when DE 38 = 0, at 28 28 28. o toto

Changing variables: the chain rule

no und variables, y... ae facina of independent variables, de

22. Changing variables in surface and volume integrals - Jacobians

[10.9 4xdy= [sus dudo where J

The Jacobian also writen as À

af du de de where now J

14. Fourier Series and Transforms

as M M oo forall points where yx) continuous

Fourier series for other ranges

Variable t range 0 < fT function of ine with period frequency wo = 25/7).

2 [ts dx
I

23. Fourier series for odd and even functions

(antisymmetric) funtion fe, y) = y) defined in the range x © x Sn only

ff stosinme ar. 1 in addition,

x = 7/2, ten the cocficients sa are given by sa = 0 (form eve)

sine terms ate required in the Fourier seis and x de in

4 [7 vs)cosms dx format,

[These results also apply to Fourier series with more general ranges provides appropriate changes are made to the
Tints of integration}

Complex form of Fourier series

ae

with tokin al integer values inthe range EM, Thi
‘conditions asthe real for.
Forotherrangesth

= Ecc, c

Variable range < Y < 1

x < me which sample inthe 2N equally spaced points 5

24. Fourier transforms

1 (a) sa function defined in the range —00 $ x co then the Rosier transfor a) is defined bythe equations

te ut at
this selairati becomes

w= fo une war

25. Convolution theorem
us) = [* stay -nar= [xt dr Sat) y) then Alu) = Ru) Hw
Parseval's theorem

[ro woa= 2 [ru ste) du (is normalised as on page 21)
Fourier transforms in two dimensions

di = [vine er

[amv te) dr azimut symmetric

Examples

o vw]

Fourier transforms in three dimensions
din = [vier ar

26. 15. Laplace Transforms

1 (tf function defined for 1 > 0, the Laplace transform

We) =cl0)= ena

“|
Leone]

Is defined by the equation

Delta fun

Unit step function

Convolution theorem

27. 16. Numerical Analysis

Finding the zeros of equations

the equations y = f(s) and x isan approximation othe oc hen either
xen =x (Norton)
orton = x (Linear interpolation)

Numerical integration of differential equations

Central difference notation
1 ys) tabulated at qual intervals of, where hs the interval then 8,2 = nd

Approximating to derivatives

CE

Interpolation: Everett's formula

wheres the fracion of the interval (= 25 — 3) between the sampling points and D = 1 — 0. The ist to

Numerical evaluation of definite integrals

The interval of integration i dived int m equal subintervals, each of width then

where = (b= @)/m ands) = a+ jh

Y integration is divided into an even number (say 2n) of equal subntras, cach of width

28. There general om f° y(n) r= Fons

17. Treatment of Random Errors
ge
ron

Range method

A quick but crude method of estimating «so find the range rf set of readings Le the difference between
thelangestand smallest vals, then

Combination of errors

1Z=Z(4,B...) (it A, B, te independent) then

twat = (Fen) + (Ge)
m 2m amas (2) =)
(iv) Z=InA, w=

29. 18, Statistics

Mean and Variance

re, the probably that Xx, le We the distribution s continuous the probability that X hes in an
interval x (2), where (3) 1 he probability density function.

Mean = E08) = Ermor fx f(s) dx.

X= Eu = Por fe wf) de

Probability distributions

a fe ep E
Weighted sums of random variables
LÉ = aX +BY then EU) = EUX) + BE(Y)- X an are independent then VI) = 2V(X) + VL

Statistics of a data sample x1...

Sample meant = À Ex

Sinn? (11%)

Regression (least squares fitting)

of points (x.y, model the observations by y = a à Blas =) +6

IE

To. straight ine by least squares to

4-150

Sample statis e

Estimates for the variances and Bare © and

Coreation cofient: À

30. -Compilation:

of 2

Mathematical

5yc/B+1=Lujm*as