Q7 7P Expand the same functions as in ... [FREE SOLUTION]

Chapter 7: Q7 7P (page 360)

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- 蟺, 蟺)$ and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Expert verified

The resultant expansion is $\frac{蟺}{4} + \frac{1}{2} ({鈭�}_{n = - 鈭�}^{鈭�} \frac{{(- 1)}^{n} - 1}{蟺 n^{2}} + \frac{i}{n} {(- i)}^{n}) e^{i n x} n 鈮� 0$ .

Step by step solution

Given data

The given function is complex exponentials $e^{i n x}$ .

Concept of Fourier series

In terms of an infinite sum of sines and cosines, a Fourier series is an expansion of a periodic function.

The orthogonality relationships of the sine and cosine functions are used in Fourier series.

Find the coefficients

The $c_{0}$ coefficient is shown below.

$c_{0} = \frac{1}{2 蟺} {鈭�}_{0}^{蟺} x d x c_{0} = \frac{蟺}{4}$

The other coefficients are as follows:

$c_{n} = \frac{1}{2 蟺} {鈭�}_{0}^{蟺} x e^{- i n x} d x a c_{n} = \frac{1}{2 蟺} {[- \frac{x}{i n} e^{- i n x} + \frac{1}{n^{2}} e^{- i n x}]}_{- 蟺}^{蟺} c_{n} = \frac{1}{2 蟺} (- \frac{蟺}{i n} e^{i n 蟺} + \frac{1}{n^{2}} (e^{- i n 蟺} -)) c_{n} = \frac{1}{2 蟺 n^{2}} ({(- 1)}^{n} + 1) + \frac{i}{2 n} (- i)^{n}$

Then the function is $\frac{蟺}{4} + \frac{1}{2} ({鈭�}_{n = - 鈭�}^{鈭�} \frac{{(- 1)}^{n} - 1}{蟺 n^{2}} + \frac{i}{n} {(- i)}^{n}) e^{i n x} n 鈮� 0$ .

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- 蟺, 蟺)$ and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Find the coefficients

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Radiation

Circular Motion and Gravitation

Dynamics

Geometrical and Physical Optics

Modern Physics

Study anywhere. Anytime. Across all devices.

91影视

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials einx on the interval(-蟺,蟺) and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Find the coefficients

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Radiation

Circular Motion and Gravitation

Dynamics

Geometrical and Physical Optics

Modern Physics

Study anywhere. Anytime. Across all devices.

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- 蟺, 蟺)$ and verify in each case that the answer is equivalent to the one found in Section 5.