fourier series function, f(x) = {x − 2} if −2 < x < 0 {x+2} if 0 < x < 2 sketch the 4 periodic extension of f and determine the fourier coefficents.l
fourier series function, f(x) = {x − 2} if −2 < x < 0 {x+2} if 0 < x < 2 sketch the 4 periodic extension of f and determine the fourier coefficents.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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Transcribed Image Text:### Fourier Series Function
#### Definition:
The function \( f(x) \) is defined piecewise as follows:
- \( f(x) = x - 2 \) for \( -2 < x < 0 \)
- \( f(x) = x + 2 \) for \( 0 < x < 2 \)
#### Task:
Sketch the 4-periodic extension of \( f \) and determine the Fourier coefficients.
### Explanation:
To complete the task, follow these steps:
1. **Sketching the Periodic Extension:**
- Extend the given piecewise function \( f(x) \) such that it repeats with a period of 4. This means replicating the piecewise segments for all intervals \( [n, n+4) \), where \( n \) is an integer.
2. **Determining the Fourier Coefficients:**
- Calculate the Fourier coefficients for the periodic function. This involves finding:
- The average value (a constant term),
- Cosine terms (representing the even harmonics), and
- Sine terms (representing the odd harmonics).
- Use the integral formulas for Fourier coefficients over one period to determine their values.
### Graph Explanation:
- No graph is provided in the image, but the expected sketch would involve repeating the linear segments \( x-2 \) and \( x+2 \) in intervals shifted by periods of 4.
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