Fourier integral of Fourier series
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Transcript of 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