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Applications of Fourier Series Calculator

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Example
Created on 2024-06-20Asked by Aiden Johnson (Solvelet student)
Find the Fourier series representation of the function f(x)=x f(x) = x on the interval [−π,π][- \pi, \pi] .

Solution

Step 1: Identify the function and the interval. The function is f(x)=x f(x) = x on the interval [−π,π][- \pi, \pi]. Step 2: Calculate the coefficients a0a_0, ana_n, and bnb_n. The Fourier series is given by: f(x)∼a02+∑n=1∞(ancos⁡(nx)+bnsin⁡(nx)). f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right). Step 3: Calculate a0a_0: a0=1π∫−ππx dx=0. a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x \, dx = 0. Step 4: Calculate ana_n: an=1π∫−ππxcos⁡(nx) dx=0. a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \cos(nx) \, dx = 0. Step 5: Calculate bnb_n: bn=1π∫−ππxsin⁡(nx) dx=2n(−1)n+1. b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \sin(nx) \, dx = \frac{2}{n} (-1)^{n+1}. Step 6: Write the Fourier series: f(x)∼∑n=1∞2n(−1)n+1sin⁡(nx). f(x) \sim \sum_{n=1}^{\infty} \frac{2}{n} (-1)^{n+1} \sin(nx). Step 7: Conclusion. The Fourier series representation of f(x)=x f(x) = x on [−π,π][- \pi, \pi] is: f(x)∼∑n=1∞2n(−1)n+1sin⁡(nx). f(x) \sim \sum_{n=1}^{\infty} \frac{2}{n} (-1)^{n+1} \sin(nx). Solved on Solvelet with Basic AI Model
Some of the related questions asked by Layla Williams on Solvelet
1. Find the Fourier series representation of the func f(x)=x f(x) = x , defined on the interval [−π,π] [-\pi, \pi] 2. Determine the convergence of the Fourier series of the func f(x)=x2 f(x) = x^2 , defined on the interval [−π,π] [-\pi, \pi]
DefinitionBasically, the Fourier series is a mathematical tool designed, in this case, for representing periodic functions as a weighted sum of elementary sine and cosine functions. There are several applications in signal processing, engineering, physics, and mathematics. This works for example to analyse/synthesise complex periodic phenomena like sound waves or electrical signals. Applications of the Fourier series include solving partial differential equations, data compression and image processing. For example, in electrical engineering, using the Aluminum series to analyze alternating current circuits and design filters to remove unwanted frequency components.
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