Successfully reported this slideshow.
Upcoming SlideShare
×

# Fourier integral of Fourier series

1,031 views

Published on

Fourier Integral of Advanced engineering Mathematics.

Published in: Engineering
• Full Name
Comment goes here.

Are you sure you want to Yes No

### Fourier integral of Fourier series

1. 1. Gandhinagar Institute ofTechnology Fourier Integral Mehta Chintan B. D1-14 3rd SEM. Mech. D Guided By:- Prof. M. S. Suthar Advanced Engineering Mathematics (2130002)
2. 2. Fourier Series β’ As we know that the fourier series of function f(x) in any interval (-l, l) is given by: β’ π π₯ = π0 + π=1 β π π cos πππ₯ πΏ + π π sin πππ₯ πΏ β’ Where:- β’ π0 = 1 2π βπ π π π‘ ππ‘ β’ π π= 1 π βπ π π π‘ πππ  πππ‘ π ππ‘ β’ π π= 1 π βπ π π π‘ π ππ πππ‘ π ππ‘
3. 3. Fourier Integral β’ Let f(x) be a function which is piecewise continuous in every finite interval in (ββ, β) and absolute integral in (ββ, β). β’ Then π π₯ = 1 π 0 β ( ββ β π π‘ πππ π π‘ β π₯ ππ‘)ππ β’ Where : β’ π = ππ π β’ π β β
4. 4. Proof of Fourier Integral π π₯ = π0 + π=1 β π π cos πππ₯ πΏ + π π sin πππ₯ πΏ π π₯ = 1 2π βπ π π π‘ ππ‘ + π=1 β 1 π βπ π π π‘ πππ  πππ‘ π πππ  πππ₯ π ππ‘ + π=1 β 1 π βπ π π π‘ π ππ πππ‘ π π ππ πππ₯ π ππ‘ = 1 2π βπ π π π‘ ππ‘ + π=1 β 1 π βπ π π(π‘) πππ  πππ‘ π πππ  πππ₯ π ππ‘ + π ππ πππ‘ π π ππ πππ₯ π ππ‘ = 1 2π βπ π π π‘ ππ‘ + 1 π π=1 β βπ π π π‘ πππ  ππ π π‘ β π₯ ππ‘
5. 5. β’ Putting π π = ππ π and βπ π = π π+1 β π π = π + 1 π π β π π = π π so βπ π π = 1 π π π₯ = 1 2π βπ π π π‘ ππ‘ + βπ π π π=1 β βπ π π π‘ πππ π π π‘ β π₯ ππ‘ β’ As π β β, 1 π = 0 and βπ π = π π β 0, the infinite series in above equation becomes an integral from 0 π‘π β π π₯ = 1 π 0 β ββ β π π‘ cos π π‘ β π₯ ππ‘ ππ β’ Now expanding πππ π(π‘ β π₯) in above equation.
6. 6. π π₯ = 1 π 0 β ( ββ β π π‘ πππ ππ‘ πππ ππ₯ + π ππππ‘ π ππππ₯) ππ = 1 π 0 β ββ β π π‘ πππ ππ‘ππ‘ πππ ππ₯ππ + 1 π 0 β ββ β π π‘ π ππππ‘ππ‘ π ππππ₯ππ = 0 β π΄ π πππ ππ₯ππ + 0 β π΅ π π ππππ₯ππ β’ Where: β’ π΄ π = 1 π ββ β π π‘ πππ ππ‘ππ‘ β’ B π = 1 π ββ β π π‘ π ππππ‘ππ‘
7. 7. Fourier cosine integrals β’ When π(π₯) is an even function: β’ π΄ π = 2 π 0 β π π‘ πππ ππ‘ππ‘ and B π = 0 β’ So the fourier integrals of an even function is given by: β’ π(π₯) = 0 β π΄ π πππ ππ₯ππ
8. 8. Fourier sin integral β’ When π(π₯) is an odd function: β’ π΄ π = 0 and B π = 2 π 0 β π π‘ π ππππ‘ππ‘ β’ So the fourier integral of odd function is given by: β’ π(π₯) = 0 β π΅ π π ππππ₯ππ
9. 9. Fourier cosine sum β’ Find the fourier cosine integral of π π = πβππ, where π > π, π > π hence show that π β πππππ π π+π π ππ = π ππ πβππ οThe fourier cosine integral of π π₯ is given by: π π₯ = 0 β π΄ π πππ ππ₯ππ π΄ π = 2 π 0 β π π‘ πππ ππ‘ππ‘ = 2 π 0 β πβππ‘ πππ ππ‘ππ‘ = 2 π πβππ‘ π2 + π2 (βππππ ππ‘ + ππ ππππ‘ (ππππ 0 π‘πβ) = 2 π ( π π2 + π2)
10. 10. β’ Hence: π π₯ = 2π π 0 β 1 π2 + π2 πππ ππ₯ππ 0 β πππ ππ₯ π2 + π2 ππ = π 2π π(π₯) = π 2π πβππ₯ (x > 0, π > 0)
11. 11. Fourier sine integral sum β’ Find the sine integral of π π₯ = πβππ₯ , hence show that π 2 πβππ₯ = 0 β ππ ππππ₯ π2+π2 ππ οThe fourier sine integral of π π₯ is given by: π(π₯) = 0 β π΅ π π ππππ₯ππ
12. 12. π΅ π = 2 π 0 β π π‘ π ππππ‘ππ‘ = 2 π 0 β πβππ‘ π ππππ‘ππ‘ = 2 π πβππ‘ π2 + π2 (βππ ππππ‘ β ππππ ππ‘) (ππππ 0 π‘π β) = 2 π ( π π2 + π2 )
13. 13. β’ Hence: π π₯ = 2 π 0 β ππ ππππ₯ π2 + π2 ππ 0 β ππ ππππ₯ π2 + π2 ππ = π 2 π π₯ 0 β ππ ππππ₯ π2 + π2 ππ = π 2 πβππ₯(x > 0, π > 0)
14. 14. References β’ Advanced engineering mathematics ofTATA McGraw Hill β’ https://www.wikipedia.org>wiki>fourier_integral β’ https://mathonline.wikidot.com
15. 15. ThankYou