find the fourier series of the function f(x)= { ((x −2≤x≤0 )),((4 0≤x≤2)) :} ? help me sir ?


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Question Number 100899 by mhmd last updated on 29/Jun/20

$f i n d t h e f o u r i e r s e r i e s o f t h e f u n c t i o n f (x) = {\begin{cases} x - 2 ⩽ x ⩽ 0 \\ 4 0 ⩽ x ⩽ 2 \end{cases} ?$ $h e l p m e s i r ?$
	Answered by bramlex last updated on 29/Jun/20

	$f (x) : o d d f u n c t i o n . L = 4$ $a_{n} = 0$ $b_{n} = \frac{2}{L} \underset{0}{\int^{L}} f (x) \sin (\frac{n π x}{L}) d x$ $b_{n} = \frac{2}{4} [\underset{- 2}{\int^{0}} x \sin (\frac{n π x}{4}) d x + \underset{0}{\int^{2}} 4 \sin (\frac{n π x}{4}) d x]$ $b_{n} = \frac{1}{2} [\frac{16}{n^{2} π^{2}} \sin (\frac{n π}{2}) + \frac{4}{n π}]$ $b_{n} = \frac{8}{{(n π)}^{2}} \sin (\frac{n π}{2}) + \frac{2}{n π}$ $f (x) = \frac{a_{0}}{2} + \underset{n = 1}{\sum^{\infty}} [\frac{8}{{(n π)}^{2}} \sin (\frac{n π}{2}) + \frac{2}{n π}] . \cos (\frac{n π x}{4})$
	Commented by mhmd last updated on 29/Jun/20

	$s i r c a n y o u s e n d t h e a l l s o l u t i o n ?$
	Commented by mhmd last updated on 29/Jun/20

	$t h a n k y o u s i r$
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