Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Finding the Fourier Series of a Piecewise Function
Consider the problem of finding the Fourier series for the given piecewise function:
\[ f(x) =
\begin{cases}
0 & \text{for } -\pi \le x < -\pi/2 \\
\sin(4x) & \text{for } -\pi/2 \le x < \pi/2 \\
0 & \text{for } \pi/2 \le x \le \pi
\end{cases}
\]
The function \(f(x)\) is defined over the interval \([-π, π]\) and is piecewise continuous. The specific intervals and corresponding function values are described as follows:
- For \( -π \le x < -π/2 \), \( f(x) = 0 \)
- For \( -π/2 \le x < π/2 \), \( f(x) = \sin(4x) \)
- For \( π/2 \le x \le π \), \( f(x) = 0 \)
To construct the Fourier series, we will need to calculate the Fourier coefficients. The Fourier series of a function \(f(x)\) defined on the interval \([-L, L]\) is given by:
\[ f(x) \sim a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right) \]
Where:
- \(a_0\) is the average value of the function over one period,
- \(a_n\) and \(b_n\) are the Fourier coefficients given by:
\[
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx
\]
\[
b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx
\]
For this specific problem, since the function is defined on \([-π, π]\), we have \(L = π\). The intervals need to be considered when setting up the integr](https://content.bartleby.com/qna-images/question/a68164dd-6bba-4aa5-92bc-4824a71db092/fb5d1a16-6dd5-4c95-be2c-0a2a0062a700/wd7c2ko_processed.jpeg)
Transcribed Image Text:### Finding the Fourier Series of a Piecewise Function
Consider the problem of finding the Fourier series for the given piecewise function:
\[ f(x) =
\begin{cases}
0 & \text{for } -\pi \le x < -\pi/2 \\
\sin(4x) & \text{for } -\pi/2 \le x < \pi/2 \\
0 & \text{for } \pi/2 \le x \le \pi
\end{cases}
\]
The function \(f(x)\) is defined over the interval \([-π, π]\) and is piecewise continuous. The specific intervals and corresponding function values are described as follows:
- For \( -π \le x < -π/2 \), \( f(x) = 0 \)
- For \( -π/2 \le x < π/2 \), \( f(x) = \sin(4x) \)
- For \( π/2 \le x \le π \), \( f(x) = 0 \)
To construct the Fourier series, we will need to calculate the Fourier coefficients. The Fourier series of a function \(f(x)\) defined on the interval \([-L, L]\) is given by:
\[ f(x) \sim a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right) \]
Where:
- \(a_0\) is the average value of the function over one period,
- \(a_n\) and \(b_n\) are the Fourier coefficients given by:
\[
a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx
\]
\[
b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx
\]
For this specific problem, since the function is defined on \([-π, π]\), we have \(L = π\). The intervals need to be considered when setting up the integr
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