fourier integral of fourier series

Gandhinagar Institute of Technology

Fourier Integral

Mehta Chintan B.D1-143rd SEM. Mech. D

Guided By:- Prof. M. S. Suthar

Advanced Engineering Mathematics (2130002)

Fourier Series

• As we know that the fourier series of function f(x) in any interval (-l, l) is given by:

• Where:-• = • =• =

Fourier Integral

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

Proof of Fourier Integral

• Putting and so

• As and the infinite series in above equation becomes an integral from

• Now expanding in above equation.

• Where:

• B

Fourier cosine integrals

• When is an even function:• and B

• So the fourier integrals of an even function is given by:

Fourier sin integral

• When is an odd function:• and B

• So the fourier integral of odd function is given by:

Fourier cosine sum• Find the fourier cosine integral of , where hence show

that The fourier cosine integral of is given by:

• Hence:

Fourier sine integral sum

• Find the sine integral of , hence show that The fourier sine integral of is given by:

• Hence:

References

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

Thank You