Fourier series

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The Fourier series of a real function $f$ defined on the interval $(-p,p)$ is given by

where \begin{eqnarray*} a_0&=&\frac{1}{p}\int_{-p}^pf(x)\,dx\\ a_n&=&\frac{1}{p}\int_{-p}^pf(x)\,\cos \left(\frac{n\, \pi}{p}x\right)\,dx\\ b_n&=&\frac{1}{p}\int_{-p}^pf(x)\,\sin \left(\frac{n\, \pi}{p}x\right)\,dx \end{eqnarray*}

Example

Consider the function defined by $f(x)=x+\pi$ for $-\pi < x < \pi $ and $f(x)=f(x+2\pi)$ for $-\infty < x < \infty$. Its Fourier series expansion is \[ f(x)=\pi+2\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n}\sin (n\,x). \]

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Simulation 1: Representation of Fourier series

The following simulation shows the partial sum of the Fourier series (up to 20 terms) for a given function $f(x)$ (defined by user).

Remark: You can observe how the partial sum converges to the given function by increasing the number of terms. Activate the box Fourier series. Use the tools Zoom In or Out, if needed.

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Other examples to try:

• $f(x)=\exp(x+1)$ as: exp(x+1). Interval: 0 < $x$ < 1.
• $f(x)=\sqrt{x-1}$ as: sqrt(x-1). Interval: -1 $\leq x$ < 0.
• $f(x)=x^2+|x|$ as: x^2+abs(x). Interval: -1 $\leq x\leq $ 1.

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Simulation 2: Activity

The following simulation shows the partial sum of the Fourier series (up to 20 terms) for a given function $f(x)$ (defined by user). However, the partial sum is NOT CORRECT, since the terms $a_0$, $a_1$ and $b_1$ are all zero by default in the simulation.

Calculate the Fourier series of the function and then type the CORRECT values of the coefficients $a_0$, $a_1$ and $b_1$ rounded to two decimal places.

For example, the coefficients for the default function $|x|+x$ are $a_0=1, a_1=-0.41$ and $b_1=0.64$. Type these values in the corresponding boxes.

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Worksheet exemplar

The following file contains activities and problems associated with the simulations.

• Fourier series approximation (pdf)
• Fourier series approximation (tex)