MATH 414 Lecture 9

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Complex Fourier Series

Example

$f(x)=x$ on interval Failed to parse (syntax error): {\displaystyle x \in ( 0, \2 \pi )}

Find complex series of form

$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{\mathrm{e}}^{inx}$

$c_n=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}f(x){\mathrm{e}}^{-inx}\mathrm{d}x=\frac{1}{2\pi }x{\mathrm{e}}^{-inx}\mathrm{d}x$

Integrating by parts gives

$\begin{array}{rl}{c}_n& =\frac{1}{2\pi }({\frac{x{\mathrm{e}}^{-inx}}{-in}|}_0^{2\pi }-\cancel{\underset{0}{\overset{2\pi }{\int }}\frac{{\mathrm{e}}^{inx}}{-in}\mathrm{d}x})\\ & =\frac{1}{-in}{\mathrm{e}}^{-2\pi in}\end{array}$

Observe that ${\mathrm{e}}^{i\theta }$ is $2\pi$-periodic, and when $\theta =2\pi n$, we get ${\mathrm{e}}^{2\pi ni}=1$. Hence we can cancel that integrated term and simplify $c_n$ to just

$c_n=\frac{i}{n}$

$c_0=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}x{\mathrm{e}}^{-i0x}\mathrm{d}x=\pi$

Therefore $f(x)=\pi +\sum _{n=-\mathrm{\infty };n\ne 0}^{\mathrm{\infty }}\frac{i{\mathrm{e}}^{inx}}{n}$

Piecewise Continuous Functions

Consider interval $x\in [a,b]$. $f$ is piecewise continuous on $[a,b]$ if and only if $f$ is continuous except for a finite number of "jump" type discontinuities.

Piecewise Smooth Functions

$f$ is piecewise smooth (pws) on $[a,b]$ if and only if $f$ is piecewise continuous on $[a,b]$ and at the jumps, both left and right derivatives exist.

The left and right derivatives of a function $f$ are defined as follows:

$\begin{array}{rl}f{\text{'}}_L(x_0)=f^{\prime }(x_0^{-})& =\underset{x\to x_0^{-}}{\lim }\left(\frac{f(x)-f(x_0^{-})}{x-x_0}\right)\\ f{\text{'}}_R(x_0)=f^{\prime }(x_0^{+})& =\underset{x\to x_0^{+}}{\lim }\left(\frac{f(x)-f(x_0^{+})}{x-x_0}\right)\end{array}$

Example: Square Wave

$f(x)=\{\begin{array}{lll}1& \text{even multiples}[0,\pi ]0& \text{odd multiples}[\pi ,2\pi ]\end{array}$

Left and right derivatives at points of discontinuities are both 0, but this does not mean the function is differentiable at $x_0$!

Theorem. Let $f(x)$ be a $2\pi$-periodic piecewise smooth function, and let $a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n\cos (nx)+b_n\sin (nx)$ be its Fourier series.

Then $\underset{N\to \mathrm{\infty }}{\lim }(a_0+\sum _{n=1}^N{a}_n\cos (nx)+b_n\sin (nx))=\{\begin{array}{lll}f(x)& f(x)\text{is point of continuity}\frac{f(x^{+})+f(x^{-})}{2}& f(x)\text{is a jump discontinuity}\end{array}$

Proof.

quod erat demonstrandum

Gibbs Phenomenon: Fourier series seems to diverge from function slightly just before function discontinuity