0, -# < x < 0, π 5. f(x)=-1, 1. 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
In each of the following problems you are given a function on the interval −π<x<π.
Sketch several periods of the corresponding periodic function of period 2π. Expand the
periodic function in a sine-cosine Fourier series.
![### Educational Website Content
#### Function Definition and Piecewise Representation
**Problem 5:**
Define the function \( f(x) \) as follows:
\[
f(x) = \begin{cases}
0, & -\pi < x < 0, \\
-1, & 0 < x < \frac{\pi}{2}, \\
1, & \frac{\pi}{2} < x < \pi.
\end{cases}
\]
This piecewise function \( f(x) \) is divided into three distinct pieces, each defined over a specific interval of \( x \):
- For \(-\pi < x < 0\), \( f(x) = 0 \).
- For \(0 < x < \frac{\pi}{2}\), \( f(x) = -1 \).
- For \(\frac{\pi}{2} < x < \pi\), \( f(x) = 1 \).
There are no graphs or diagrams accompanying this function in the current context. If a visual representation were provided, it would typically depict three horizontal line segments on the \( x \)-axis corresponding to the values of \( f(x) \) over each interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5846127e-0595-4419-aab6-08da9e640d47%2F8f38ffa1-72eb-4ae7-bb73-cf9816d372d6%2Fl2rxn21_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
#### Function Definition and Piecewise Representation
**Problem 5:**
Define the function \( f(x) \) as follows:
\[
f(x) = \begin{cases}
0, & -\pi < x < 0, \\
-1, & 0 < x < \frac{\pi}{2}, \\
1, & \frac{\pi}{2} < x < \pi.
\end{cases}
\]
This piecewise function \( f(x) \) is divided into three distinct pieces, each defined over a specific interval of \( x \):
- For \(-\pi < x < 0\), \( f(x) = 0 \).
- For \(0 < x < \frac{\pi}{2}\), \( f(x) = -1 \).
- For \(\frac{\pi}{2} < x < \pi\), \( f(x) = 1 \).
There are no graphs or diagrams accompanying this function in the current context. If a visual representation were provided, it would typically depict three horizontal line segments on the \( x \)-axis corresponding to the values of \( f(x) \) over each interval.
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