Problem Set 2: Fourier series on different intervals

1. Fourier series of $x\sin(px)$

Calculate the Fourier series of $f(x) = x\sin(px)$ on $x \in [-\pi, \pi]$ and where $p$ is a positive integer. Use this result to calculate values of the sums, \[ \sum_{n=2}^\infty \frac{(-1)^n}{n^2-1}\qquad \text{and} \qquad \sum_{n=3}^\infty \frac{1}{n^2-4}. \]

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2. Fourier series of even extension of $\sin x$

Calculate the Fourier series for the even, periodic extension of $f(x) = \sin x$ for $x\in [0, \pi]$. Use this to deduce that \[ \sum_{n=1}^\infty \frac{1}{4n^2 - 1} = \frac{1}{2}. \]

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3. Fourier series over alternative regions

b. i) Choosing the periodic extension

Suppose that the function $f(x)$ is defined only on $[0, L]$, Show that we can write \begin{align*} f(x) &= \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos \left(\frac{n\pi x}{L}\right) \\ f(x) &= \sum_{n=1}^\infty b_n \sin \left(\frac{n\pi x}{L}\right) \\ f(x) &= \sum_{n=1}^\infty c_n \sin \left(\frac{[2n-1]\pi x}{2L}\right) \end{align*}

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b. ii) Different approximations for $f(x) \equiv 1$

Find $a_n$, $b_n$, and $c_n$ for $f(x) \equiv 1$ on $[0, L]$.

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4. Fourier series of neither even-nor-odd extension

Calculate the Fourier series of the periodic function $f$ of period $2L$ defined by \[ f(x) = \begin{cases} x + L & -L < x \leq 0, \\ 0 & 0 < x \leq L. \end{cases} \]

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5. Complex Fourier Series

Consider the complex Fourier series of the form \[ f(x) \sim \sum_{n=-\infty}^\infty c_n e^{inx}, \]

which can also be used to represent functions of period $2\pi$.

a) Derivation of the Fourier coefficients

Using the standard definitions for a Fourier series expansion, derive the expressions for the complex Fourier series coefficients.

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b) Derivative of the series

If $f(x) = e^x$ for $-\pi < x < \pi$ with Fourier such that \[ f(x) = \sum_{n=-\infty}^\infty c_n e^{inx}, \qquad |x| < \pi. \]

If we differentiate this equation, we have \[ e^x = \sum_{n=-\infty}^\infty (in) c_n e^{inx} \Rightarrow c_n = (in) c_n, \]

and hence $(1-in)c_n = 0$ so $c_n = 0$ for all $n$. Where is the mistake?

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{PHT, Lincoln Maths, 2012-2013, www.theshapeofmath.com/oxford/maths/fourier}