Use Residus theorem to prove that ∀ a>0 Σ_(n=0) ^∞ (1/( n^2 +a^2 )) = (1/2)((π/(ash(πa))) −(1/a^2 )) and Σ_(n=0) ^∞ (((−1)^n )/(n^2 +a^2 )) = (1/2)((( π)/(a.th(πa))) −(1/a^2 )) Assume that we can developp in integer serie the functions f(x)=(x/(shx)) and g(x)=(x/(thx)) Give the DL_2 of f and g around zero Why can′t we use that theorem to explicit f(a)=Σ_(n=0) ^∞ (((−1)^n )/( (2n+1)^2 +a^2 )) ???


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Question Number 69236 by ~ À ® @ 237 ~ last updated on 21/Sep/19

$U s e R e s i d u s t h e o r e m t o p r o v e t h a t \forall a > 0 \underset{n = 0}{\sum^{\infty}} \frac{1}{n^{2} + a^{2}} = \frac{1}{2} (\frac{π}{a s h (π a)} - \frac{1}{a^{2}})$ $a n d \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n^{2} + a^{2}} = \frac{1}{2} (\frac{π}{a . t h (π a)} - \frac{1}{a^{2}})$ $A s s u m e t h a t w e c a n d e v e l o p p i n i n t e g e r s e r i e t h e f u n c t i o n s$ $f (x) = \frac{x}{s h x} a n d g (x) = \frac{x}{t h x}$ $G i v e t h e {D L}_{2} o f f a n d g a r o u n d z e r o$ $W h y {c a n}^{'} t w e u s e t h a t t h e o r e m t o e x p l i c i t$ $f (a) = \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(2 n + 1)}^{2} + a^{2}} ? ? ?$
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