MATH 414 Lecture 8

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Sine and Cosine Series

A regular Fourier series expresses a function in terms of both sines and cosines. For example, on the 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, \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_{k=1}^\infty a_k \, \cos{(k\,x)} + b_k \sin{(k\,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}}

Sine Series

The Fourier Sine Series 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(x)} on an 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 \left[ 0, \pi \right]} is a fourier series that produces an odd extension of the 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(x)} (i.e. for 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 < x < 0} , 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)} .)

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) = \sum_{k=1}^\infty b_k \, \sin{(k \, x)} b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \, \sin{(n \, x)} \,\mathrm{d}x \end{align}}

Cosine Series

The Fourier Cosine Series 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(x)} on an interval $\left[0,\pi \right]$ is a fourier series that produces an even extension of the 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(x)} (i.e. for 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 < x < 0} , 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)} .

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_{k=1}^\infty a_k \, \cos{(k \, x)} \\ a_0 = \frac{1}{\pi} \, \int_{0}^{\pi} f(x) \,\mathrm{d}x \\ a_n = \frac{2}{\pi} \, \int_{0}^{\pi} f(x) \, \cos{(n\,x)} \,\mathrm{d}x \end{align} \,\!}

Complex Fourier Series

Recall

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} \mathrm{e}^{i \, n \, x} &= \cos{(n \, x)} + i \, \sin{(n \, x)} \\ \mathrm{e}^{-i \, n \, x} &= \cos{(n \, x)} - i \, \sin{(n \, x)} \end{align}}

Let's add and divide by 2:

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} \cos{(n \, x)} &= \frac{1}{2} \left( \mathrm{e}^{i \, n \, x} + \mathrm{e}^{-i \, n \, x} \right) \\ \sin{(n \, x)} &= \frac{1}{2i} \left( \mathrm{e}^{i \, n \, x} + \mathrm{e}^{-i \, n \, x} \right) \end{align}}

Substituting this into the fourier series definition and simplifying gives

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_{k=1}^\infty \left( \frac{a_k - i \, b_k}{2} \right) \mathrm{e}^{i \, n \, x} + \sum_{k=1}^\infty \left( \frac{a_k + i \, b_n}{2} \, \mathrm{e}^{-i \, n \, x} \right) \,\!}

We'll rewrite this formula a different way. 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 c_0 := a_0} , 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_k := \frac{a_k - i \, b_k}{2}} . If we change the index on our sum by negating, we get

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) = \sum_{n=0}^{\infty} c_n \, \mathrm{e}^{i \, n \, x} + \sum_{n=-\infty}^-1 \left( \frac{a_{-n} + i \, b_{-n}}{2} \right) \mathrm{e}^{i \, n \, x}}

Observe that 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 \left( \frac{a_{-n} + i \, b_{-n}}{2} \right)} is the same as 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_n} , so we're left with

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) &= \sum_{n=-\infty}^\infty c_n \, \mathrm{e}^{i \, n \, x} \\ c_0 &= \frac{1}{2\pi} \, \int_{-\pi}^{\pi} f(x) \, \mathrm{e}^{-0x} \,\mathrm{d}x = \frac{1}{2 \, \pi} \, \int_{-\pi}^{\pi} f(x) \,\mathrm{d}x \\ c_n &= \frac{1}{2\pi} \, \int{-\pi}^{\pi} f(x) \, \mathrm{e}^{-i \, n \, x} \,\mathrm{d}x \end{align}}