MATH 414 Lecture 7

« previous | Wednesday, January 29, 2014 | next »

Last Time

Standard Interval

$-\pi \leq x\leq \pi$

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

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 -c \le x \le c}

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

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{\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{align}}

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

In this case, 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 = 2} , so

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 = \frac{1}{2 \cdot 2} \int_{-2}^{2} f(x) \,\mathrm{d}x = \frac{1}{4}}

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}{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) }}

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}{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

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) = \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 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 function. 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 \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, 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 \int_{-\pi+c}^{\pi+c} F(x) \, \mathrm{d}x} is independent of 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 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}}