Q3P Sketch several periods of the co... [FREE SOLUTION]

Chapter 7: Q3P (page 354)

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

Short Answer

Expert verified

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

Step by step solution

Given

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

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

The Fourier series for the function :

$f (x) = \frac{a_{0}}{2} + {鈭�}_{n = 1}^{鈭�} (a_{n} \cos nx + b_{n} \sin nx) a_{0} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) d x a_{n} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) c o s n x d x b_{n} = \frac{1}{蟺} {鈭�}_{- 蟺}^{蟺} f (x) s i n n x d x$

If is an even function:

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

If is an odd function:

$a_{0} = a_{a} = 0 f (x) = {鈭�}_{n = 1}^{鈭�} b_{n} \sin nx$

From the given information

The sketch of the given function is shown below.

Coefficients of :

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

Coefficients of :

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

Calculate Coefficients of bn

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

Hence the expansion is

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

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91影视

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

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

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91影视

Sketch several periods of the corresponding periodic function of period . Expand the periodic2蟺 function in a sine-cosine Fourier series.f(x)={0,-蟺<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

Fields in Physics

Electricity and Magnetism

Kinematics Physics

Quantum Physics

Rotational Dynamics

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

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