If f(x) is an even function, i.e. f(-x) = f(x), then its Fourier series reduces to the Fourier cosine series. f(x )-ao + E an cos (Ex) E an COS 1 ao L f(x)dx 2 (п
If f(x) is an even function, i.e. f(-x) = f(x), then its Fourier series reduces to the Fourier cosine series. f(x )-ao + E an cos (Ex) E an COS 1 ao L f(x)dx 2 (п
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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![If \( f(x) \) is an even function, i.e., \( f(-x) = f(x) \), then its Fourier series reduces to the Fourier cosine series.
\[
f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos \left( \frac{n \pi}{L} x \right)
\]
\[
a_0 = \frac{1}{L} \int_0^L f(x) \, dx
\]
\[
a_n = \frac{2}{L} \int_0^L f(x) \cos \left( \frac{n \pi}{L} x \right) dx \quad \text{for} \quad n = 1, 2, 3, \ldots
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2140facb-0981-40cd-bf23-81e224855fc4%2Fbb369a90-a514-4a19-8f5b-e2a060761d52%2Fo6mfrx_processed.png&w=3840&q=75)
Transcribed Image Text:If \( f(x) \) is an even function, i.e., \( f(-x) = f(x) \), then its Fourier series reduces to the Fourier cosine series.
\[
f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos \left( \frac{n \pi}{L} x \right)
\]
\[
a_0 = \frac{1}{L} \int_0^L f(x) \, dx
\]
\[
a_n = \frac{2}{L} \int_0^L f(x) \cos \left( \frac{n \pi}{L} x \right) dx \quad \text{for} \quad n = 1, 2, 3, \ldots
\]
Transcribed Image Text:**Problem 12.3.15:** Expand \( f(x) = x^2 \), \(-1 < x < 1\) into an appropriate cosine or sine series.
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