Chapter 7: Q13P (page 358)

Repeat the example using the same Fourier series but at $x = π / 2$ .

Short Answer

Expert verified

At, $x = \frac{π}{2}$ :

$f (\frac{1}{2}) = (\begin{matrix} \frac{1}{4} + \frac{1}{π} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{π} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{π}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

Step by step solution

Given

The given function is $f (x) = \{\begin{cases} 0, - π < x < 0 \\ \sin x, 0 < x < π \end{cases}$

The given point is $x = \frac{π}{2}$ .

Definition of Fourier series

The Fourier series for the function $f (x)$ :

$f (x) = \frac{a_{0}}{2} + \sum_{n - 1}^{\infty} (a_{n} \cos nx + b_{n} sinx) a_{0} = \frac{1}{π} \int_{- x}^{x} f (x) d x a_{n} = \frac{1}{π} \int_{- x}^{x} f (x) c o s n x d x b_{n} = \frac{1}{π} \int_{- x}^{x} f (x) s i n n x d x$

If $f (x)$ is an even function:

$b_{n} = 0 f (x) = \frac{a_{0}}{2} + \sum_{n - 1}^{\infty} a_{n} c o s n x$

If $f (x)$ is an odd function:

$a_{0} = a_{n} = 0 f (x) = \sum_{n - 1}^{\infty} b_{n} s i n n x$

Write the Fourier series of the function and input the value

The Fourier series of function is shown below.

$f (x) - (\begin{matrix} \frac{1}{4} + \frac{1}{π} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{π} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{π}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

Now, at $x - \frac{π}{2}$ .

role="math" localid="1659238212014" $f (\frac{1}{2}) - (\begin{matrix} \frac{1}{4} + \frac{1}{π} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{π} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{π}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

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.

Repeat the example using the same Fourier series but at $x = π / 2$ .

Short Answer

Step by step solution

Given

Definition of Fourier series

Write the Fourier series of the function and input the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Modern Physics

Classical Mechanics

Geometrical and Physical Optics

Circular Motion and Gravitation

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

Repeat the example using the same Fourier series but at x=π/2.

Short Answer

Step by step solution

Given

Definition of Fourier series

Write the Fourier series of the function and input the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Modern Physics

Classical Mechanics

Geometrical and Physical Optics

Circular Motion and Gravitation

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

Repeat the example using the same Fourier series but at $x = π / 2$ .