Suppose that the Fourier series for a periodic function f(x) on the interval (-1, ) is given as f(x) = 2/3 +3¹-( n=1 Then, the value of the integral [ -T [1- (-1)"] n² f(x) cos(4x) dx = Select one: -cos(nx) + n=1 (-1)" 2n -sin(nx).
Suppose that the Fourier series for a periodic function f(x) on the interval (-1, ) is given as f(x) = 2/3 +3¹-( n=1 Then, the value of the integral [ -T [1- (-1)"] n² f(x) cos(4x) dx = Select one: -cos(nx) + n=1 (-1)" 2n -sin(nx).
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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Mm.3
![**Understanding Fourier Series and Integral Calculations**
Consider a periodic function \( f(x) \) defined on the interval \((- \pi, \pi)\). The Fourier series representation of \( f(x) \) is given by:
\[ f(x) = \frac{3}{8} + 3 \sum_{n=1}^{\infty} \frac{[1 - (-1)^n]}{n^2} \cos(nx) + \sum_{n=1}^{\infty} \frac{(-1)^n}{2n} \sin(nx). \]
We are interested in evaluating the following integral:
\[ \int_{-\pi}^{\pi} f(x) \cos(4x) \, dx. \]
**Select one of the following options as the value of the integral:**
- \(\frac{3}{8\pi}\)
- \(\frac{3}{8} \pi\)
- \(\frac{3}{8}\)
- \(\frac{1}{8}\)
**Explanation:**
1. **Fourier Series Analysis:**
The Fourier series decomposes \( f(x) \) into a sum of sines and cosines. Each term in the series has a specific coefficient that weighs the contribution of different frequency components to the function \( f(x) \).
2. **Integral Calculation:**
To solve the integral \( \int_{-\pi}^{\pi} f(x) \cos(4x) \, dx \), we leverage the orthogonality properties of the sine and cosine functions. Specifically, integrals of products of different frequency cosine functions over a complete period (here \( -\pi \) to \( \pi \)) result in zero unless the frequencies match exactly.
3. **Evaluating the Given Choices:**
After thorough analysis, we conclude that the correct value of the integral is:
\[ \boxed{\frac{3}{8}} \]
This value corresponds to the option circled in the provided choices.
Understanding this result requires comprehending Fourier series and orthogonal functions in detail, which are fundamental concepts in mathematical analysis and signal processing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F816cc47c-f059-4914-940d-8196a7c7af6b%2Fc06fcffa-cbb4-489e-ab3d-4b8b86ec1f5b%2Fv6cmquo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Fourier Series and Integral Calculations**
Consider a periodic function \( f(x) \) defined on the interval \((- \pi, \pi)\). The Fourier series representation of \( f(x) \) is given by:
\[ f(x) = \frac{3}{8} + 3 \sum_{n=1}^{\infty} \frac{[1 - (-1)^n]}{n^2} \cos(nx) + \sum_{n=1}^{\infty} \frac{(-1)^n}{2n} \sin(nx). \]
We are interested in evaluating the following integral:
\[ \int_{-\pi}^{\pi} f(x) \cos(4x) \, dx. \]
**Select one of the following options as the value of the integral:**
- \(\frac{3}{8\pi}\)
- \(\frac{3}{8} \pi\)
- \(\frac{3}{8}\)
- \(\frac{1}{8}\)
**Explanation:**
1. **Fourier Series Analysis:**
The Fourier series decomposes \( f(x) \) into a sum of sines and cosines. Each term in the series has a specific coefficient that weighs the contribution of different frequency components to the function \( f(x) \).
2. **Integral Calculation:**
To solve the integral \( \int_{-\pi}^{\pi} f(x) \cos(4x) \, dx \), we leverage the orthogonality properties of the sine and cosine functions. Specifically, integrals of products of different frequency cosine functions over a complete period (here \( -\pi \) to \( \pi \)) result in zero unless the frequencies match exactly.
3. **Evaluating the Given Choices:**
After thorough analysis, we conclude that the correct value of the integral is:
\[ \boxed{\frac{3}{8}} \]
This value corresponds to the option circled in the provided choices.
Understanding this result requires comprehending Fourier series and orthogonal functions in detail, which are fundamental concepts in mathematical analysis and signal processing.
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