Laplace transform & fourier series

It includes laplace transform and fourier series explanation and its properties.with some solved examples.

1. Laplace Transform Definition, Standard transforms & properties By VAIBHAV TAILOR

2. The Laplace transform of a function, f ( t ), is defined as where F ( s ) is the symbol for the Laplace transform, L is the Laplace transform operator, and f ( t ) is some function of time, t . Definition of Laplace Transform

3. Sr. No Function f(t ) Laplace Transformation L(f) 1. 1 2. 3. 4. Standard Laplace Transformations

4. Sr. No Function f(t) Laplace Transformation L(f) 5. 6. 7. Standard Laplace Transformations

5. By definition, the inverse Laplace transform operator ,L - 1 , converts an s -domain function back to the corresponding time domain function Inverse Laplace Transform

6. No . S domain function T domain function 1 2 3 4 5 6 7 Inverse Laplace Transformations

7. Linearity property Properties of Laplace Transform

8. First S hifting property

9. Multiplication by t

10. Division by t

11. Example Convolution Theorem

12. Convolution Theorem

13. Fourier series Definition, formula, odd & even function, Half range series & some example

14. A Fourier series may be defined as an expansion of a function in a series of sines and cosines such as , Henceforth we assume f satisfies the following ( Dirichlet ) conditions: f(x ) is a periodic function ; f(x) has only a finite number of finite discontinuities; f(x ) has only a finite number of extrem values, maxima and minima in the interval [0,2p ]. Definition

15. The formula for a Fourier series is We have formulae for the coefficients (for the derivations see the course notes): FORMULA

16. Even Functions The value of the function would be the same when we walk equal distances along the X-axis in opposite directions . Mathematically speaking Even and Odd Functions

17. Odd Functions The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions . Mathematically speaking - Even and Odd Functions

18. The Fourier series of an even function is expressed in terms of a cosine series . The Fourier series of an odd function is expressed in terms of a sine series. Half Range Series

19. Fourier coefficients function even function odd function neither Summary of finding coefficients

20. Fourier series expansion with period Find the fourier series of f(x)=x in interval Solution The fourier series of f(x)with period is given by

21. Put in equation 1

22. Find fourier series of The fourier series of with period is given by Fourier series expansion with Arbitrary function

23. Find the fourier series of in the interval is an even function here, The fourier series of an even function with period is given by Odd even

24. Find Fourier sine series of in interval Here, ; The fourier sine series of in interval Half range sine series

25. Find Fourier cosine series of in rang Here, ; The Fourier cosine series of in interval is Where Half range cosine series

26. Thank you