Chapter 10.3, Problem 36E

Complex Form of the Fourier Series.

a. Using the Euler formula $e^{i\theta }=\cos \theta +i\sin \theta ,i=\sqrt{-1},$ prove that

$\cos nx=\frac{e^{inx}+e^{-inx}}{2}$ and $\sin nx=\frac{e^{inx}-e^{-inx}}{2i}$.

b. Show that the Fourier series

$\begin{array}{rll}f\left(x\right)& \sim & \frac{a_0}{2}+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\left\{a_n\cos nx+b_n\sin nx\right\}}\\ & =& c_0+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\left\{c_n{e}^{inx}+c_{-n}{e}^{-inx}\right\}},\end{array}$

where

$c_0=\frac{a_0}{2},c_n=\frac{a_n-i{b}_n}{2},c_{-n}=\frac{a_n+i{b}_n}{2}.$

c. Finally, use the results of part (b) to show that

$f\left(x\right)\sim {\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum }}{c}_n{e}^{inx},-\pi <x<\pi }$

where

$c_n=\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }f\left(x\right)e^{-inx}dx}$.