Chapter 2.4, Problem 2.19P

(a)

To determine

The proof that $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\left[a_n\sin \left(\frac{n\pi x}{a}\right)+b_n\cos \left(\frac{n\pi x}{a}\right)\right]}$ equivalent to ${\displaystyle \sum _{n=-\infty }^{\infty }{c}_n{e}^{in\pi x/a}}$ and also the value of $c_n$.

(b)

To determine

The proof that $c_n=\frac{1}{2a}{\displaystyle {\int }_{-a}^{+a}f\left(x\right)e^{-in\pi x/a}dx}$.

(c)

To determine

The proof that (a) and (b) becomes $f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle \sum _{n=-\infty }^{\infty }F\left(k\right)e^{ikx}\Delta k}$ and $f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-a}^{+a}f\left(x\right)e^{-ikx}dx}$.

(d)

To determine

The Plancherel’s theorem for $a\to \infty$.