f(x) = { ₁ Sketch the odd periodic extension of f. 1 0
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Mathematics and Fourier Series: Periodic Extensions and Fourier Series**
Consider the function \( f(x) \) defined by:
\[ f(x) = \begin{cases}
1 & 0 < x < \pi/2 \\
2 & \pi/2 < x < \pi
\end{cases} \]
**Tasks:**
1. (a)-(c) [redacted]
2. (d) Sketch the odd periodic extension of \( f \).
**Fourier Series Analysis:**
(e) **Find the Fourier sine series of \( f \)**.
(f) **Determine the convergence of the Fourier sine series**:
- At \( x = 0 \)
- At \( x = \pi/2 \)
- At \( x = \pi \)
- At \( x = 3\pi/2 \)
- At \( x = 2\pi \)
(g) **Even Periodic Extensions**:
- Denote by \( f_{ep}(x) \) the even periodic extension of \( f(x) \).
- Consider when we use periodic functions of the following form to approximate \( f_{ep}(x) \):
\[ T(x) = A_0 + A_1 \cos x + B_1 \sin x + A_2 \cos(2x) + B_2 \sin(2x) \]
- The error in mean is defined by:
\[ \int_{-\pi}^{\pi} |f_{ep}(x) - T(x)|^2 \, dx \]
- Determine the coefficients \( A_0, A_1, B_1, A_2, B_2 \) that minimize the error in mean.
This transcription provides detailed instructions for students to perform tasks related to Fourier series, focusing on periodic extensions, convergence, and approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F17e6b4b5-4b21-4b4d-a932-a5673d242802%2Fvhqumwm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics and Fourier Series: Periodic Extensions and Fourier Series**
Consider the function \( f(x) \) defined by:
\[ f(x) = \begin{cases}
1 & 0 < x < \pi/2 \\
2 & \pi/2 < x < \pi
\end{cases} \]
**Tasks:**
1. (a)-(c) [redacted]
2. (d) Sketch the odd periodic extension of \( f \).
**Fourier Series Analysis:**
(e) **Find the Fourier sine series of \( f \)**.
(f) **Determine the convergence of the Fourier sine series**:
- At \( x = 0 \)
- At \( x = \pi/2 \)
- At \( x = \pi \)
- At \( x = 3\pi/2 \)
- At \( x = 2\pi \)
(g) **Even Periodic Extensions**:
- Denote by \( f_{ep}(x) \) the even periodic extension of \( f(x) \).
- Consider when we use periodic functions of the following form to approximate \( f_{ep}(x) \):
\[ T(x) = A_0 + A_1 \cos x + B_1 \sin x + A_2 \cos(2x) + B_2 \sin(2x) \]
- The error in mean is defined by:
\[ \int_{-\pi}^{\pi} |f_{ep}(x) - T(x)|^2 \, dx \]
- Determine the coefficients \( A_0, A_1, B_1, A_2, B_2 \) that minimize the error in mean.
This transcription provides detailed instructions for students to perform tasks related to Fourier series, focusing on periodic extensions, convergence, and approximation.
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