Problem 4 f(x) = an = 1 if x < 4/6 3 if Find the Fourier cosine series of f 2 (3 An (NTT) - 2 Din (NT)). MIT Note: 4164x24/2 else where The even extension of f to [-L₁ L] is if 02x2L 3f1-x) if -L
Problem 4 f(x) = an = 1 if x < 4/6 3 if Find the Fourier cosine series of f 2 (3 An (NTT) - 2 Din (NT)). MIT Note: 4164x24/2 else where The even extension of f to [-L₁ L] is if 02x2L 3f1-x) if -L
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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Question
4
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![### Problem 4
Given the piecewise function \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
1 & \text{if } x < L/6 \\
3 & \text{if } L/6 < x < L/2 \\
0 & \text{elsewhere}
\end{cases}
\]
Find the Fourier Cosine Series of \( f \).
### Solution:
The coefficient \( A_n \) is given by:
\[
A_n = \frac{2}{n\pi} \left( 3 \sin \left( \frac{n\pi}{2} \right) - 2 \sin \left( \frac{n\pi}{6} \right) \right)
\]
### Notes:
- The even extension of \( f \) to the interval \([-L, L]\) is:
\[
F(x) =
\begin{cases}
f(x) & \text{if } 0 < x < L \\
f(-x) & \text{if } -L < x < 0
\end{cases}
\]
- The odd extension of \( f \) to the interval \([-L, L]\) is:
\[
F(x) =
\begin{cases}
f(x) & \text{if } 0 < x < L \\
-f(-x) & \text{if } -L < x < 0
\end{cases}
\]
This problem focuses on finding the Fourier series representation of a given piecewise function, highlighting the distinction between even and odd extensions to expand the function over a given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Feb9d2a05-b319-4718-b097-8e98466a162b%2Fwo23kz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4
Given the piecewise function \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
1 & \text{if } x < L/6 \\
3 & \text{if } L/6 < x < L/2 \\
0 & \text{elsewhere}
\end{cases}
\]
Find the Fourier Cosine Series of \( f \).
### Solution:
The coefficient \( A_n \) is given by:
\[
A_n = \frac{2}{n\pi} \left( 3 \sin \left( \frac{n\pi}{2} \right) - 2 \sin \left( \frac{n\pi}{6} \right) \right)
\]
### Notes:
- The even extension of \( f \) to the interval \([-L, L]\) is:
\[
F(x) =
\begin{cases}
f(x) & \text{if } 0 < x < L \\
f(-x) & \text{if } -L < x < 0
\end{cases}
\]
- The odd extension of \( f \) to the interval \([-L, L]\) is:
\[
F(x) =
\begin{cases}
f(x) & \text{if } 0 < x < L \\
-f(-x) & \text{if } -L < x < 0
\end{cases}
\]
This problem focuses on finding the Fourier series representation of a given piecewise function, highlighting the distinction between even and odd extensions to expand the function over a given interval.
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