Fourier integral of Fourier series
-
Upload
chintan-mehta -
Category
Engineering
-
view
283 -
download
13
Transcript of Fourier integral of Fourier series
![Page 1: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/1.jpg)
Gandhinagar Institute of Technology
Fourier Integral
Mehta Chintan B.D1-143rd SEM. Mech. D
Guided By:- Prof. M. S. Suthar
Advanced Engineering Mathematics (2130002)
![Page 2: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/2.jpg)
Fourier Series
• As we know that the fourier series of function f(x) in any interval (-l, l) is given by:
• Where:-• = • =• =
![Page 3: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/3.jpg)
Fourier Integral
• Let f(x) be a function which is piecewise continuous in every finite interval in () and absolute integral in ().• Then • Where :
![Page 4: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/4.jpg)
Proof of Fourier Integral
![Page 5: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/5.jpg)
• Putting and so
• As and the infinite series in above equation becomes an integral from
• Now expanding in above equation.
![Page 6: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/6.jpg)
• Where:
• B
![Page 7: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/7.jpg)
Fourier cosine integrals
• When is an even function:• and B
• So the fourier integrals of an even function is given by:
![Page 8: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/8.jpg)
Fourier sin integral
• When is an odd function:• and B
• So the fourier integral of odd function is given by:
![Page 9: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/9.jpg)
Fourier cosine sum• Find the fourier cosine integral of , where hence show
that The fourier cosine integral of is given by:
![Page 10: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/10.jpg)
• Hence:
![Page 11: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/11.jpg)
Fourier sine integral sum
• Find the sine integral of , hence show that The fourier sine integral of is given by:
![Page 12: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/12.jpg)
![Page 13: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/13.jpg)
• Hence:
![Page 14: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/14.jpg)
References
• Advanced engineering mathematics of TATA McGraw Hill• https://www.wikipedia.org>wiki>fourier_integral • https://mathonline.wikidot.com
![Page 15: Fourier integral of Fourier series](https://reader035.fdocuments.net/reader035/viewer/2022081417/587234011a28ab102f8b45f3/html5/thumbnails/15.jpg)
Thank You