Oblain Fourier series for the function $f(x)=\pi x, \quad 0 \leq x \leq 1$ $=\pi(2-x), \quad 1 \leq x \leq 2 .$

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written 3.0 years ago by

prajapatijaimin • 3.3k

Solution:

Let, $f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos n \pi x+\sum_{n=1}^{\infty} b_{n} \sin u \pi x \\$

Then, $a_{0}=\int_{0}^{2} f(x) d x=\int_{0}^{1} \pi x d x+\int_{1}^{2} \pi(2-x) d x=\pi\left[\frac{x^{2}}{2}\right]{0}^{1}+\pi\left[2 x-\frac{x^{2}}{2}\right]{1}^{2} \\$

$=\pi\left(\frac{1}{2}\right)+\pi\left[(4-2)-\left(2-\frac{1}{2}\right)\right]=\pi \\$

$$ \begin{aligned} a_{n} &=\int_{0}^{2} f(x) \cos n \pi x d x=\int_{0}^{1} \pi x \cos …

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