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Problem 15 Consider the function of with period 4 and geven on the interval [-2₁2] hy x f(x) = 3₁ if if -15x²1 The Fourier Semes of f is of 15x28 if -22×2-1 ( (-2² conit + 4 omnil) pm ni x 2 π7²n² n=1 a) can you use the Fouver series off to cleduce the Fouver series of f'? explais use the Fouer series of f to deduce the termer series of 6) Can you the anti derivative F(x) = √² ft) dt 7 c) Find the Founess seves of F. 2 Ans: Fl=+(+²) (²2₂) EX es 6 ทะ 4 om

Problem 15 Consider the function of with period 4 and geven on the interval [-2₁2] hy x f(x) = 3₁ if if -15x²1 The Fourier Semes of f is of 15x28 if -22×2-1 ( (-2² conit + 4 omnil) pm ni x 2 π7²n² n=1 a) can you use the Fouver series off to cleduce the Fouver series of f'? explais use the Fouer series of f to deduce the termer series of 6) Can you the anti derivative F(x) = √² ft) dt 7 c) Find the Founess seves of F. 2 Ans: Fl=+(+²) (²2₂) EX es 6 ทะ 4 om

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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**Problem 15** Consider the function \( f \) with period 4 and given on the interval \([-2, 2]\) by \[ f(x) = \begin{cases} x & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } 1 \leq x < 2 \\ -1 & \text{if } -2 < x \leq -1 \end{cases} \] The Fourier series of \( f \) is \[ \sum_{n=1}^{\infty} \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} + \frac{4}{\pi^2 n^2} \sin \frac{n\pi}{2} \right) \sin \frac{n\pi}{2} x \] a) Can you use the Fourier series of \( f \) to deduce the Fourier series of \( f' \)? Explain. b) Can you use the Fourier series of \( f \) to deduce the Fourier series of the antiderivative \[ F(x) = \int_{0}^{x} f(t) \, dt \] c) Find the Fourier series of \( F \). **Answer:** \[ F(x) = \frac{7}{6} + \sum_{n=1}^{\infty} \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} + \frac{4}{\pi^2 n^2} \sin \frac{n\pi}{2} \right) \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} \right) \]
Transcribed Image Text:**Problem 15** Consider the function \( f \) with period 4 and given on the interval \([-2, 2]\) by \[ f(x) = \begin{cases} x & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } 1 \leq x < 2 \\ -1 & \text{if } -2 < x \leq -1 \end{cases} \] The Fourier series of \( f \) is \[ \sum_{n=1}^{\infty} \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} + \frac{4}{\pi^2 n^2} \sin \frac{n\pi}{2} \right) \sin \frac{n\pi}{2} x \] a) Can you use the Fourier series of \( f \) to deduce the Fourier series of \( f' \)? Explain. b) Can you use the Fourier series of \( f \) to deduce the Fourier series of the antiderivative \[ F(x) = \int_{0}^{x} f(t) \, dt \] c) Find the Fourier series of \( F \). **Answer:** \[ F(x) = \frac{7}{6} + \sum_{n=1}^{\infty} \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} + \frac{4}{\pi^2 n^2} \sin \frac{n\pi}{2} \right) \left( -\frac{2}{n\pi} \cos \frac{n\pi}{2} \right) \]
Expert Solution
Step 1

Given: A periodic function f(x) with a fundamental period of X=4 can be written as for period [-2, 2]

fx=-1,-2≤x≤-1x,-1≤x≤11,1≤x≤2.

 

The trigonometric Fourier series of the function f(x) is also given as

fx=∑n=1∞-2nπcosnπ2+4n2π2sinnπ2sinnπx2   ...1.

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