Q5P Sketch several periods of the co... [FREE SOLUTION]

Chapter 7: Q5P (page 355)

Sketch several periods of the corresponding periodic function of period $2 蟿蟿$ . Expand the periodic function in a sine-cosine Fourier series.
role="math" localid="1659236419546" $f (x) = {\begin{matrix} 0, & - 蟺 < x < 0 \\ - 1, & 0 < x < \frac{蟺}{2} \\ 1, & \frac{蟺}{2} < x < 蟺 \end{matrix}$

Short Answer

Expert verified

The answer of the given function is

$f (x) = - \frac{2}{蟺} [{鈭�}_{n = 1 + 4 m}^{鈭�} \frac{\cos n x}{n} - {鈭�}_{n = 1 + 4 m}^{鈭�} \frac{\cos n x}{n}] - \frac{4}{蟺} [{鈭�}_{n = 1 + 4 m}^{鈭�} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2 . . . .$

Step by step solution

Given

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

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

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

$f (x) = \frac{a_{0}}{2} + {鈭�}_{n = 1}^{鈭�} (a_{n} \cos nx + b_{n} \sin nx) a_{0} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) dx a_{0} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) \cos nxdx b_{n} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) \sin nxdx If f (x) is an even function : b_{n} = 0 f (x) = \frac{a_{0}}{2} + {鈭�}_{n = 1}^{鈭�} a_{n} \cos nx If f (x) is an odd function : a_{0} = a_{n} = 0 f (x) = {鈭�}_{n = 1}^{鈭�} b_{n} \sin nx$

From the given information

The sketch of the given function shown below.

Coefficients of $a_{n}$ :

$a_{n} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) \cos nxdx = \frac{1}{蟺} [- {鈭�}_{0}^{\frac{蟺}{2}} cosnxdx + {鈭�}_{\frac{蟺}{2}}^{蟺} cosnxdx] = \frac{1}{苍蟺} [- {[\sin nx]}^{\frac{蟺}{2}} 0 + {[\sin nx]}_{\frac{蟺}{2}}^{蟺}] = - \frac{2}{苍蟺} \sin (\frac{苍蟺}{2})$

Calculate Coefficients of bn

Coefficients of $b_{n}$ are given below.

$b_{n} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) \sin n x d x = \frac{1}{蟺} [- {鈭�}_{0}^{\frac{蟺}{2}} s in n x d x + {鈭�}_{\frac{蟺}{2}}^{蟺} \sin n x d x] = \frac{1}{苍蟺} [{[\cos n x]}_{0}^{\frac{蟺}{2}} - {[\cos n x]}_{\frac{蟺}{2}}^{蟺}] = \frac{1}{苍蟺} [2 \cos (\frac{n 蟺}{2}) - \{1 + {(- 1)}^{n}\}] H e n c e t h e e x p a n s i o n i s f (x) = - \frac{2}{蟺} [{鈭�}_{n = 1 + 4 m}^{鈭�} \frac{\cos n x}{n} - {鈭�}_{n = 3 + 4 m}^{鈭�} \frac{\cos n x}{n}] - \frac{4}{蟺} [{鈭�}_{n = 2 + 4 m}^{鈭�} \frac{\sin n x}{n}] w h e r e m = 0, 1, 2 - -$

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.

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

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

Force

Work Energy and Power

Torque and Rotational Motion

Thermodynamics

Electricity

Linear Momentum

Study anywhere. Anytime. Across all devices.

91影视

Sketch several periods of the corresponding periodic function of period 2蟿蟿 . Expand the periodic function in a sine-cosine Fourier series.role="math" localid="1659236419546" f(x)={0,-蟺<x<0-1,0<x<蟺21,蟺2<x<蟺

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

Force

Work Energy and Power

Torque and Rotational Motion

Thermodynamics

Electricity

Linear Momentum

Study anywhere. Anytime. Across all devices.