Chapter 7.8, Problem 10P

(a) Sketch several periods of the function $f(x)$ of period 2 $\pi$ which is equal to $x$ on $-\pi <x<\pi .$ Expand $f(x)$ in a sine-cosine Fourier series and in a complex exponential Fourier series.

Answer: $f(x)=2\left(\sin x-\frac{1}{2}\sin 2x+\frac{1}{3}\sin 3x-\frac{1}{4}\sin 4x+\cdots \right)$.

(b) Sketch several periods of the function $f(x)$ of period 2 $\pi$ which is equal to $x$ on $0<x<2\pi .$ Expand $f(x)$ in a sine-cosine Fourier series and in a complex exponential Fourier series. Note that this is not the same function or the same series as (a).