Advance Engineering Mathematics Formulas

Engineering Mathematics Formulas

1. X =0 gives the Maclaurin Series

2. ⇨ S in0 = 0 ; first term is x Cos0 = 1 ; first term is 1 L n and exp cancel each other

3. s inh(-x) = -sinh(x); This is ODD function c osh(-x) = conh(x); This is EVEN function Derivative of sinh(x) and cosh(x) s inh(2x) = 2sinh(x)*cosh(x) C osh(2x) = 2cosh 2 (x)-1

4. Complex Number C omplex conjugate Absolute value of complex number Absolute value of complex conjugate number are same F or second quadrant, slope is minus C o mplex number in Polar co-ordinate

5. Complex Exponential Function w here

6. Trigonometric Function y = sin(x) & cos(x) Eg.

7. v -> nyu differential

8. ← Bessel function of the first kind of order 0 ← Modified Bessel function of the first kind of order 0 R emember values of J1/2 and J(-1/2)

9. First kind or second kind ← generalization for 1st/2nd kind not for the modified bessel function + This is the product rule of derivative - sum subtract

10. 1st/2nd kind modified the re is no –ve here

11. T(1) = 1 T(2) = 2 T(3) = T(2+1) = 2T(2) E rf(x) is 0 to x E rfc(X) is x to infinity

12. Differential Equations M ultiply both side by integrating factor Standard Form M(x,y)dx + N(x,y)dy = 0

13. Second Order Differential Equations Linear differential operator

14. Second Order Differential Equations

15. Second Order Differential Equations If the Wronskian is zero then the functions are linearly dependent.

16. Second Order Differential Equations E xample:

17. Second Order Differential Equations

18. Second Order Differential Equations is solution

19. Euler-Cauchy Differential Equations

20. w rto. z w rto. x

21. Examples

22. Examples Particular solution in next slides

23. Particular solution C omplementary solution makes ODE = 0 Y p is particular solution

24. Lp annihilates f(x) C1*exp(x)+C2*exp(-x)+C3*exp(-x) P articular solution

25. solution

26. Electric Circuit See example of a parallel circuit

27. Complex Analysis of a Simple Electric Circuits I f a=0, phase angle = pi/2

28. W =2*pi*frequency(Hz)

29. V rms = V peak /sqrt(2)

30. Y 1, y2, y3 are two time dependent variables like pressure, velocity, temperature…

31. C omplementary + Particular solution S olution of homogeneous differential equation

32. Example: s ame as E igen values matrix E igen vector matrix

33. Orthogonal function M atlab code

34. zero

35. a = avg value of the function a

36. E xpressing ω in terms of T T = 2 in this case

37. Laplace Transform

38. ETC