[3] Find the Fourier series of the function given by f(x) = { f -a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Fourier Series Calculation and Graph Analysis
#### Problem Statement
**3.** Find the Fourier series of the function given by:
\[
f(x) = \begin{cases}
0 & \text{if } -a < x < 0, \\
3x & \text{if } 0 < x < a.
\end{cases}
\]
Sketch the graph of \( f(x) \) and its periodic extension. To what values does the Fourier series converge at \( x = -a \), \( x = -a/2 \), \( x = 0 \), \( x = a/2 \), \( x = a \), and \( x = 2a \)?
#### Explanation
##### Fourier Series
The Fourier series of a function \( f(x) \) is an expansion of the function in terms of sine and cosine functions. For a periodic function \( f(x) \) with period \( 2L \), the Fourier series is given by:
\[
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{n\pi x}{L} \right) + b_n \sin\left( \frac{n\pi x}{L} \right) \right),
\]
where the coefficients \( a_0 \), \( a_n \), and \( b_n \) are calculated as follows:
\[
a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx,
\]
\[
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n\pi x}{L} \right) \, dx,
\]
\[
b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n\pi x}{L} \right) \, dx.
\]
##### Graph Description
To sketch the function \( f(x) \) and its periodic extension, follow these steps:
1. For \( x \) in the interval \( -a < x < 0 \), the function \( f(x) \) is zero. This describes a horizontal line along the x-axis.
2. For](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F910d9f6f-8821-4234-89ce-92a603013dee%2Fceauge7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Fourier Series Calculation and Graph Analysis
#### Problem Statement
**3.** Find the Fourier series of the function given by:
\[
f(x) = \begin{cases}
0 & \text{if } -a < x < 0, \\
3x & \text{if } 0 < x < a.
\end{cases}
\]
Sketch the graph of \( f(x) \) and its periodic extension. To what values does the Fourier series converge at \( x = -a \), \( x = -a/2 \), \( x = 0 \), \( x = a/2 \), \( x = a \), and \( x = 2a \)?
#### Explanation
##### Fourier Series
The Fourier series of a function \( f(x) \) is an expansion of the function in terms of sine and cosine functions. For a periodic function \( f(x) \) with period \( 2L \), the Fourier series is given by:
\[
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{n\pi x}{L} \right) + b_n \sin\left( \frac{n\pi x}{L} \right) \right),
\]
where the coefficients \( a_0 \), \( a_n \), and \( b_n \) are calculated as follows:
\[
a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx,
\]
\[
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n\pi x}{L} \right) \, dx,
\]
\[
b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n\pi x}{L} \right) \, dx.
\]
##### Graph Description
To sketch the function \( f(x) \) and its periodic extension, follow these steps:
1. For \( x \) in the interval \( -a < x < 0 \), the function \( f(x) \) is zero. This describes a horizontal line along the x-axis.
2. For
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