Chapter 7: Q3P (page 354)

Sketch several periods of the corresponding periodic function of period . Expand the periodic $2 π$ function in a sine-cosine Fourier series.
$f (x) = {\begin{cases} 0, & - π & < & x & < & \frac{π}{2} \\ 1, & \frac{π}{2} & < & x & < & π, \end{cases}$

Short Answer

Expert verified

The expansion is $[\sum_{n = 1 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 3 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 1 + 2 m}^{\infty} \frac{\sin n x}{n} + 2 \sum_{n = 2 + 4 m}^{\infty} \frac{\sin n x}{n}]$ $w h e r e m = 0, 1, 2, - - .$

Step by step solution

Given

The given function is $f (x) = {\begin{cases} 0, & - π & < & x & < & 0 \\ 1, & 0 & < & x & < & \frac{π}{2}, \\ 0, & \frac{π}{2} & < & x & < & π . \end{cases}$ .

The concept of the Fourier series for the function f(x)

The Fourier series for the function :

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

If is an even function:

$b_{n} = 0 f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} a_{n} \cos nx$

If is an odd function:

$a_{0} = a_{a} = 0 f (x) = \sum_{n = 1}^{\infty} b_{n} \sin nx$

From the given information

The sketch of the given function is shown below.

Coefficients of :

$a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) d x = \frac{1}{π} \int_{- π}^{\frac{π}{2}} d x = \frac{1}{π} {[x]}_{0}^{\frac{π}{2}} = \frac{1}{2}$

Coefficients of :

$a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) s i n n x d x = \frac{1}{π} \int_{0}^{\frac{π}{2}} f (x) s i n d x = \frac{1}{π} {[\cos d x]}_{0}^{\frac{π}{2}} = \frac{1}{π} [1 - \cos (\frac{n π}{2})]$

Calculate Coefficients of bn

Coefficients of $b_{n}$ are $\frac{1}{π}, \frac{2}{2 π}, \frac{1}{3 π}, 0, \frac{1}{5 π}, \frac{2}{6 π}, \frac{1}{7 π}$ .

Hence the expansion is

$f (x) = \frac{1}{4} + \frac{1}{π} [\sum_{n = 1 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 3 + 4}^{\infty} \frac{\cos n x}{n} - \sum_{n = 1 + 2 m}^{\infty} \frac{\sin n x}{n} + 2 \sum_{n = 2 + 4 m}^{\infty} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2, - - .$

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.

Sketch several periods of the corresponding periodic function of period . Expand the periodic $2 π$ function in a sine-cosine Fourier series.
$f (x) = {\begin{cases} 0, & - π & < & x & < & \frac{π}{2} \\ 1, & \frac{π}{2} & < & x & < & π, \end{cases}$

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function f(x)

From the given information

Calculate Coefficients of bn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Famous Physicists

Force

Translational Dynamics

Medical Physics

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Sketch several periods of the corresponding periodic function of period . Expand the periodic2π function in a sine-cosine Fourier series.f(x)={0,-π<x<π21,π2<x<π,

Short Answer

Step by step solution

Given

The concept of the Fourier series for the function f(x)

From the given information

Calculate Coefficients of bn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Famous Physicists

Force

Translational Dynamics

Medical Physics

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Sketch several periods of the corresponding periodic function of period . Expand the periodic $2 π$ function in a sine-cosine Fourier series.
$f (x) = {\begin{cases} 0, & - π & < & x & < & \frac{π}{2} \\ 1, & \frac{π}{2} & < & x & < & π, \end{cases}$