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 −L≤x≤L. Graph f for Exercises 8.2.2, 8.2.6, 8.2.8, 8.2.15, and 8.2.16 and Fm(x)=a0+m∑n=1(ancosnπxL+bnsinnπxL) on the same axes for variable values of m.
2. L=1; f(x)=2−x
3. L=π; f(x)=2x−3x2
4. L=1; f(x)=1−3x2
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<0,1−x2,−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.
- Find the Fourier series of f(x)=ex on [−π,π].
- Deduce from (a) that ∞∑n=01n2+1=πcothπ−12.
21. Find the Fourier series of f(x)=(x−π)cosx on [−π,π].
22. Find the Fourier series of f(x)=(x−π)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.
- 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),−L≤x≤L, with a0=12L∫L−Lf(x)dx, an=−Ln2π2∫L−Lf″(x)cosnπxLdx,andbn=−Ln2π2∫L−Lf″(x)sinnπxLdx,n≥1.
- Show that if, in addition to the assumptions in (a), f″ is continuous and f‴ is piecewise continuous on [−L,L], then an=L2n3π3∫L−Lf‴(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=1L∫L0f(x)dx, and An=2L∫L0f(x)cos2nπxLdx,Bn=2L∫L0f(x)sin2nπxLdx,n=1,2,3,….


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(2n−1)πxL+Bnsin(2n−1)πxL),
where
An=2L∫L0f(x)cos(2n−1)πxLdxandBn=2L∫L0f(x)sin(2n−1)πxLdx,n=1,2,3,….