MATH 414 Lecture 7

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Standard 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 -\pi \le x \le \pi}

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(x) &= a_0 + \sum_{n=1}^\infty a_n \, \cos{(n \, x)} + b_n \, \sin{(n \, x)} \\ a_0 &= \frac{1}{2\pi} \int_{-\pi}^\pi f(x) \,\mathrm{d}x \\ a_n &= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, \cos{(n \, x)} \,\mathrm{d}x \\ b_n &= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, \sin{(n \, x)} \,\mathrm{d}x \end{align}}

Scaled Interval

$-c\leq x\leq c$

If we let $t={\frac {\pi }{c}}\,x$, then $x={\frac {c}{\pi }}$, and evaluating our expressions with respect to $t$ yields.

${\begin{aligned}f(x)&=a_{0}+\sum _{n=1}^{\infty }a_{n}\,\cos {\left({\frac {n\,\pi }{c}}\,x\right)}+b_{n}\,\sin {\left({\frac {n\,\pi }{c}}\,x\right)}\\a_{0}&={\frac {1}{2a}}\int _{-a}^{a}f(x)\,\mathrm {d} x\\a_{n}&={\frac {1}{a}}\int _{-a}^{a}f(x)\,\cos {\left({\frac {n\,\pi }{a}}\,x\right)}\,\mathrm {d} x\\b_{n}&={\frac {1}{a}}\int _{-a}^{a}f(x)\,\sin {\left({\frac {n\,\pi }{a}}\,x\right)}\,\mathrm {d} x\end{aligned}}$

Example

$f(x)={\begin{cases}1&0\leq x\leq 1\\0&{\mbox{otherwise}}\end{cases}}\quad \quad x\in \left[-2,2\right]$

In this case, $a=2$, so

$a_{0}={\frac {1}{2\cdot 2}}\int _{-2}^{2}f(x)\,\mathrm {d} x={\frac {1}{4}}$

$a_{n}={\frac {1}{2}}\int _{-2}^{2}f(x)\,\cos {\left({\frac {n\,\pi \,x}{2}}\right)}\,\mathrm {d} x={\frac {1}{2}}\int _{0}^{1}\,\cos {\left({\frac {n\,\pi \,x}{2}}\right)}\,\mathrm {d} x={\frac {1}{n\,\pi }}\,\sin {\left({\frac {n\,\pi }{2}}\right)}$

$b_{n}={\frac {1}{2}}\int _{-2}^{2}f(x)\,\sin {\left({\frac {n\,\pi \,x}{2}}\right)}\,\mathrm {d} x={\frac {1}{2}}\int _{0}^{1}\,\sin {\left({\frac {n\,\pi \,x}{2}}\right)}\,\mathrm {d} x={\frac {1}{n\,\pi }}\,\left(1-\cos {\left({\frac {n\,\pi }{2}}\right)}\right)$

Hence

$f(x)={\frac {1}{4}}+\sum _{n=1}^{\infty }{\frac {\sin {\left({\frac {n\,\pi }{2}}\right)}}{n\,pi}}\,\cos {\left({\frac {n\,\pi \,x}{2}}\right)}+{\frac {1-\cos {\left({\frac {n\,\pi }{2}}\right)}}{n\,\pi }}\,\sin {\left({\frac {n\,\pi \,x}{2}}\right)}$

Shifted Interval

Lemma. Let $F(x)$ be a $2\pi$-periodic function. Then

$\int _{-\pi +c}^{\pi +c}F(x)\,\mathrm {d} x=\int _{-\pi }^{\pi }F(x)\,\mathrm {d} x=\int _{0}^{\pi }F(x)\,\mathrm {d} x\,\!$

In other words, $\int _{-\pi +c}^{\pi +c}F(x)\,\mathrm {d} x$ is independent of $c$.

Proof. 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 G(c) = \int_{-\pi+c}^{\pi+c} F(x) \,\mathrm{d}x} .

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} \frac{\mathrm{d}G}{\mathrm{d}c} &= \frac{\mathrm{d}}{\mathrm{d}c} \left( \int_{0}^{\pi+c} F(x) \,\mathrm{d}x \right) - \frac{\mathrm{d}}{\mathrm{d}c} \left( \int_{0}^{c-\pi} F(x) \,\mathrm{d}x \right) \\ &= F(\pi + c) - F(c - \pi) \\ &= F(\pi + c) - f(c + \pi) \\ &= 0 \end{align}}

quod erat demonstrandum

Periodic Extension

Suppose 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)} is defined 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 0 \le x \le 2 \pi} and undefined elsewhere.

A periodic extension basically copies the defined part of the function into the undefined part

If the period of a function is 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 T} , 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 f(t) = a_0 + \sum_{n=1}^\infty a_n \, \cos{ \left( \frac{2 \pi \, n \, t}{T} \right)} + b_n \, \sin{ \left( \frac{2\pi \, n \, t}{T} \right)}}
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_n = \frac{1}{ \left( \frac{T}{2} \right)} \int_{0}^{T} f(t) \, \cos{ \left( \frac{2 \pi \, n \, t}{n} \right)}}
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 b_n = \frac{1}{ \left( \frac{T}{2} \right)} \int_{0}^{T} f(t) \, \sin{ \left( \frac{2 \pi \, n \, t}{n} \right)}}

Example

For example, take 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}

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 f(x) = a_0 + \sum_{n=1}^\infty a_n \, \cos{n \, x} + b_n \, \sin{n \, x}}

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 b_n = \frac{1}{\pi} \int_{0}^{2\pi} x \, \sin{n \, x} \,\mathrm{d}x}

etc...

In the end, we have 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) = \pi - \sum_{n=1}^\infty \frac{2 \, \sin{n \, x}}{n}}