Fourier integral of Fourier series

Fourier Integral of Advanced engineering Mathematics.

1. Gandhinagar Institute of Technology Fourier Integral Mehta Chintan B. D1-14 3 rd SEM. Mech. D Guided By:- Prof. M. S. Suthar Advanced E ngineering Mathematics ( 2130002)

2. Fourier Series As we know that the fourier series of function f(x) in any interval (-l, l) is given by: Where:- = = =  

3. Fourier Integral Let f(x) be a function which is piecewise continuous in every finite interval in ( ) and absolute integral in ( ). Then Where :  

4. Proof of Fourier Integral  

5. Putting and so As and the infinite series in above equation becomes an integral from Now expanding in above equation.  

6. Where: B  

7. Fourier cosine integrals When is an even function: and B So the fourier integrals of an even function is given by:  

8. Fourier sin integral When is an odd function: and B So the fourier integral of odd function is given by:  

9. Fourier cosine sum Find the fourier cosine integral of , where hence show that The fourier cosine integral of is given by:  

10. Hence:  

11. Fourier sine integral sum Find the sine integral of , hence show that The fourier sine integral of is given by:  

13. Hence:  

14. References Advanced engineering mathematics of TATA McGraw Hill https://www.wikipedia.org>wiki>fourier_integral https://mathonline.wikidot.com

15. Thank You