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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
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
### 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.
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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