Q10P In each of the following problem... [FREE SOLUTION]

Chapter 7: Q10P (page 355)

In each of the following problems you are given a function on the interval $- 蟺 < x < 蟺$ .Sketch several periods of the corresponding periodic function of period $2 蟿蟿$ . Expand the periodic function in a sine-cosine Fourier series,
$f (x) = {\begin{cases} - x, - 蟺 < x < 0 \\ x, 0 < x < 蟺 \end{cases}$

Short Answer

Expert verified

The resultant expansion is $f (x) = \frac{蟺}{2} + \frac{4}{蟺} {鈭�}_{k = 2 n + 1}^{鈭�} \frac{\cos (k x)}{k^{2}} n = 0, 1, 2, 3, . . . .$ .

Step by step solution

Given data

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

Concept of Fourier series

An infinite sum of sines and cosines is used to represent the expansion of a periodic function f(x) into a Fourier series.

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

Sketch the points

The sketch for the function is shown below.

Step 4: Find the coefficients of the series

The function is even, so $B_{n} = 0$ .

The rest of the coefficients are given below.

$A_{0} = \frac{2}{蟺} {鈭�}_{0}^{蟺} (蟺 - x) d x = \frac{2}{蟺} {[蟺虫 - \frac{1}{2} x^{2}]}_{0}^{蟺} A_{0} = 蟺 A_{0} = \frac{2}{蟺} {鈭�}_{0}^{蟺} (蟺 - x) \cos (nx)$

Calculate further as follows:

$= \frac{2}{蟺} {[\frac{蟺}{n} \sin (n x) - \frac{x}{n} \sin (n x) - \frac{1}{n^{2}} \cos (n x)]}_{0}^{蟺} A_{n} = - \frac{2}{{蟺苍}^{2}} ({(- 1)}^{n} - 1) = \frac{4}{{蟺办}^{2}}$

Where $k = 2 n + 1$ and $n = 0, 1, 2, 3, . . . .$ .

Thus, the function is $f (x) = \frac{蟺}{2} + \frac{4}{蟺} {鈭�}_{k = 2 n + 1}^{鈭�} \frac{\cos (k x)}{k^{2}} n = 0, 1, 2, 3, . . .$ .

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Short Answer

Step by step solution

Given data

Concept of Fourier series

Sketch the points

Step 4: Find the coefficients of the series

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In each of the following problems you are given a function on the interval -蟺<x<蟺 .Sketch several periods of the corresponding periodic function of period 2蟿蟿 . Expand the periodic function in a sine-cosine Fourier series,f(x)={-x,-蟺<x<0x,0<x<蟺

Short Answer

Step by step solution

Given data

Concept of Fourier series

Sketch the points

Step 4: Find the coefficients of the series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Further Mechanics and Thermal Physics

Medical Physics

Circular Motion and Gravitation

Classical Mechanics

Radiation

Study anywhere. Anytime. Across all devices.