Chapter 7: Q7MP (page 387)

Given $f (x) = | x |$ on $(- π, π)$ , expand $f (x)$ in an appropriate Fourier series of period $2 π$ .

Short Answer

Expert verified

With given function $f (x) = | x |$ on interval $(- π, π)$ , an appropriate Fourier series of period $2 π$ is:

$f (x) = \frac{π}{2} - \frac{4}{π} \sum_{n o d d}^{\infty} \frac{\cos (n x)}{n^{2}}$

Step by step solution

Meaning of the Fourier series

A Fourier series is defined as an infinite series used in the analysis of periodic functions, in which the terms are constants multiplied by sine or cosine functions of integer multiples of the variable.

Given parameter

Given the function $f (x) = | x |$ and interval $(- π, π)$ .

Expanding the function

The function $f (x) = | x |$ is even about Zero.

This means $b_{n} = 0$ .

As a result,

$\begin{matrix} a_{0} = \frac{2}{π} \int_{0}^{π} x d x \\ = \frac{2}{π} \frac{1}{2} π^{2} \\ = π \end{matrix}$

$\begin{matrix} a_{n} = \frac{2}{π} \int_{0}^{π} x \cos (n x) d x \\ = {\frac{2}{π} [\frac{x}{n} \sin (n x) + \frac{1}{n^{2}} \cos (n x)] |}_{0}^{π} \\ = \frac{2}{n^{2} π} ({(- 1)}^{n} - 1) \\ = - \frac{1}{n π^{2}} \end{matrix}$

Note: n is odd

Then the function will be:

$f (x) = \frac{π}{2} - \frac{4}{π} \sum_{n o d d}^{\infty} \frac{\cos (n x)}{n^{2}}$

So, the Fourier series will be $f (x) = \frac{π}{2} - \frac{4}{π} \sum_{n o d d}^{\infty} \frac{\cos (n x)}{n^{2}}$ .

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Given $f (x) = | x |$ on $(- π, π)$ , expand $f (x)$ in an appropriate Fourier series of period $2 π$ .

Short Answer

Step by step solution

Meaning of the Fourier series

Given parameter

Expanding the function

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Given f(x)=|x|on (-π,π), expand f(x)in an appropriate Fourier series of period2π.

Short Answer

Step by step solution

Meaning of the Fourier series

Given parameter

Expanding the function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Geometrical and Physical Optics

Space Physics

Electrostatics

Linear Momentum

Dynamics

Study anywhere. Anytime. Across all devices.

Given $f (x) = | x |$ on $(- π, π)$ , expand $f (x)$ in an appropriate Fourier series of period $2 π$ .