MATH 414 Lecture 9

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Complex Fourier Series

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x} on interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in ( 0, \2 \pi )}

Find complex series of form

$f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,\mathrm {e} ^{i\,n\,x}$

$c_{n}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(x)\,\mathrm {e} ^{-i\,n\,x}\,\mathrm {d} x={\frac {1}{2\pi }}x\,\mathrm {e} ^{-i\,n\,x}\,\mathrm {d} x$

Integrating by parts gives

${\begin{aligned}c_{n}&={\frac {1}{2\pi }}\left(\left.{\frac {x\,\mathrm {e} ^{-i\,n\,x}}{-i\,n}}\right|_{0}^{2\pi }-{\cancel {\int _{0}^{2\pi }{\frac {\mathrm {e} ^{i\,n\,x}}{-i\,n}}\,\mathrm {d} x}}\right)\\&={\frac {1}{-i\,n}}\,\mathrm {e} ^{-2\pi \,i\,n}\end{aligned}}$

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

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

$c_{0}={\frac {1}{2\pi }}\int _{0}^{2\pi }x\,\mathrm {e} ^{-i\,0\,x}\,\mathrm {d} x=\pi$

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

Piecewise Continuous Functions

Consider interval $x\in \left[a,b\right]$. $f$ is piecewise continuous on $[a,b]$ if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous except for a finite number of "jump" type discontinuities.

Piecewise Smooth Functions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is piecewise smooth (pws) on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is piecewise continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and at the jumps, both left and right derivatives exist.

The left and right derivatives of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} are defined as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f'_L(x_0) = f'(x_0^-) &= \lim_{x \to x_0^-} \left( \frac{f(x) - f(x_0^-)}{x - x_0} \right) \\ f'_R(x_0) = f'(x_0^+) &= \lim_{x \to x_0^+} \left( \frac{f(x) - f(x_0^+)}{x - x_0} \right) \end{align}}

Example: Square Wave

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} 1 & \mbox{even multiples}\ [0, \pi] 0 & \mbox{odd multiples}\ [\pi, 2\pi] \end{cases}}

Left and right derivatives at points of discontinuities are both 0, but this does not mean the function is differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} !

Theorem. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} be a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi} -periodic piecewise smooth function, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 + \sum_{n=1}^\infty a_n \, \cos{(n\,x)} + b_n \, \sin{(n \, x)}} be its Fourier series.

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} \left( a_0 + \sum_{n=1}^N a_n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)} \right) = \begin{cases} f(x) & f(x)\ \mbox{is point of continuity} \frac{f(x^+) + f(x^-)}{2} & f(x)\ \mbox{is a jump discontinuity} \end{cases}}

Proof.

quod erat demonstrandum

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