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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Waves Physics

Magnetism and Electromagnetic Induction

Engineering Physics

Electricity and Magnetism

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

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

Linear Momentum

Waves Physics

Magnetism and Electromagnetic Induction

Engineering Physics

Electricity and Magnetism

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

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