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rayman123
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1. Homework Statement [/b
expand the function in Fourier series[tex]f(x)= x^2+xcosx[/tex]
I divided the function in separate part and try to expand it in Fourier series separately
i have started with
[tex]x^2[/tex]
[tex]a_{0}= \frac{1}{2\pi}\int_{-\infty}^{\infty}x^2dx=\frac{\pi^2}{3}[/tex]
[tex]a_{n}= \frac{2}{\pi}\int_{0}^{\pi}x^2cosnxdx=\frac{2}{\pi}[\frac{x^2}{n}sinnx-2\int_{0}^{\pi}x\frac{sinx}{n}]dx=\frac{2}{\pi}[\frac{2\pi(-1)^n}{n^2}]=4\cdot\frac{(-1)^n}{n^2}\Rightarrow x^2= 4 \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}cosnx[/tex]
[tex]x^2[/tex] if even function then[tex]b_{n}=0[/tex]
function f(x)=x is an odd function [tex]a_{n}= 0[/tex]
[tex]b_{n}= \frac{2}{\pi}\int_{0}^{\pi}xsinnxdx=\frac{2}{\pi}[-\frac{x}{n}cosnx+\frac{1}{n}\int_{0}^{\pi}cosnxdx]=\frac{2}{\pi}[-\frac{\pi}{n}(-1)^n]=2 \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}sinnx[/tex]
is this correct so far?
can someone help me and show me how to calculate the rest? I mean how to connect everything together with cosx?
expand the function in Fourier series[tex]f(x)= x^2+xcosx[/tex]
I divided the function in separate part and try to expand it in Fourier series separately
i have started with
[tex]x^2[/tex]
[tex]a_{0}= \frac{1}{2\pi}\int_{-\infty}^{\infty}x^2dx=\frac{\pi^2}{3}[/tex]
[tex]a_{n}= \frac{2}{\pi}\int_{0}^{\pi}x^2cosnxdx=\frac{2}{\pi}[\frac{x^2}{n}sinnx-2\int_{0}^{\pi}x\frac{sinx}{n}]dx=\frac{2}{\pi}[\frac{2\pi(-1)^n}{n^2}]=4\cdot\frac{(-1)^n}{n^2}\Rightarrow x^2= 4 \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}cosnx[/tex]
[tex]x^2[/tex] if even function then[tex]b_{n}=0[/tex]
function f(x)=x is an odd function [tex]a_{n}= 0[/tex]
[tex]b_{n}= \frac{2}{\pi}\int_{0}^{\pi}xsinnxdx=\frac{2}{\pi}[-\frac{x}{n}cosnx+\frac{1}{n}\int_{0}^{\pi}cosnxdx]=\frac{2}{\pi}[-\frac{\pi}{n}(-1)^n]=2 \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}sinnx[/tex]
is this correct so far?
can someone help me and show me how to calculate the rest? I mean how to connect everything together with cosx?
Homework Equations
The Attempt at a Solution
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