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Mathematics LibreTexts

8.2E: Fourier Series I (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Q8.2.1

1. Prove Theorem 8.1.5.

Q8.2.2

In Exercises 8.2.2-8.2.16 find the Fourier series of f on [L,L] and determine its sum for LxL. Graph f for Exercises 8.2.2, 8.2.6, 8.2.8, 8.2.15, and 8.2.16 and Fm(x)=a0+mn=1(ancosnπxL+bnsinnπxL) on the same axes for variable values of m.

2. L=1; f(x)=2x

3. L=π; f(x)=2x3x2

4. L=1; f(x)=13x2

5. L=π; f(x)=|sinx|

6. L=π; f(x)=xcosx

7. L=π; f(x)=|x|cosx

8. L=π; f(x)=xsinx

9. L=π; f(x)=|x|sinx

10. L=1; f(x)={0,1<x<12,cosπx,12<x<12,0,12<x<1

11. L=1; f(x)={0,1<x<12,xcosπx,12<x<12,0,12<x<1

12. L=1; f(x)={0,1<x<12,sinπx,12<x<12,0,12<x<1

13. L=1; f(x)={0,1<x<12,|sinπx|,12<x<12,0,12<x<1

14. L=1; f(x)={0,1<x<12,xsinπx,12<x<12,0,12<x<1

15. L=4; f(x)={0,4<x<0,x,0<x<4

16. L=1; f(x)={x2,1<x<0,1x2,0<x<1

Q8.2.3

17. Verify the Gibbs phenomenon for f(x)={2,2<x<1,1,1<x<1,1,1<x<2.

18. Verify the Gibbs phenomenon for f(x)={2,3<x<2,3,2<x<2,1,2<x<3.

19. Deduce from Example 11.2.5 that

n=01(2n+1)2=π28.

20.

  1. Find the Fourier series of f(x)=ex on [π,π].
  2. Deduce from (a) that n=01n2+1=πcothπ12.

21. Find the Fourier series of f(x)=(xπ)cosx on [π,π].

22. Find the Fourier series of f(x)=(xπ)sinx on [π,π].

23. Find the Fourier series of f(x)=sinkx (k integer) on [π,π].

24. Find the Fourier series of f(x)=coskx (k integer) on [π,π].

25.

  1. Suppose f(L)=f(L), f(L)=f(L), f is continuous, and f is piecewise continuous on [L,L]. Use Theorem 8.2.4 and integration by parts to show that f(x)=a0+n=1(ancosnπxL+bnsinnπxL),LxL, with a0=12LLLf(x)dx, an=Ln2π2LLf(x)cosnπxLdx,andbn=Ln2π2LLf(x)sinnπxLdx,n1.
  2. Show that if, in addition to the assumptions in (a), f is continuous and f is piecewise continuous on [L,L], then an=L2n3π3LLf(x)sinnπxLdx.

26. Show that if f is integrable on [L,L] and f(x+L)=f(x),L<x<0 (Figure 8.2.8), then the Fourier series of f on [L,L] has the form A0+n=1(Ancos2nπL+Bnsin2nπL) where A0=1LL0f(x)dx, and An=2LL0f(x)cos2nπxLdx,Bn=2LL0f(x)sin2nπxLdx,n=1,2,3,.

clipboard_e8b9d8d7cff94b10352d35e1242f0fe57.png
Figure 8.2.8: y=f(x), where f(x+L)=f(x),L<x<0
clipboard_eb01c4d2a59759a9b9b608fb3a8baf868.png
Figure 8.2.9: y=f(x), where f(x+L)=f(x),L<x<0

27. Show that if f is integrable on [L,L] and

f(x+L)=f(x),L<x<0

(Figure 8.2.9), then the Fourier series of f on [L,L] has the form

n=1(Ancos(2n1)πxL+Bnsin(2n1)πxL),

where

An=2LL0f(x)cos(2n1)πxLdxandBn=2LL0f(x)sin(2n1)πxLdx,n=1,2,3,.

 


This page titled 8.2E: Fourier Series I (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Zoya Kravets.

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