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Use-Residus-theorem-to-prove-that-a-gt-0-n-0-1-n-2-a-2-1-2-pi-ash-pia-1-a-2-and-n-0-1-n-n-2-a-2-1-2-pi-a-th-pia-1-a-2-Assume




Question Number 69236 by ~ À ® @ 237 ~ last updated on 21/Sep/19
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 ))   ???
$${Use}\:\:{Residus}\:{theorem}\:{to}\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi}{{ash}\left(\pi{a}\right)}\:\:\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$${and}\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\:\pi}{{a}.{th}\left(\pi{a}\right)}\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$$\:\:\:{Assume}\:{that}\:{we}\:{can}\:{developp}\:{in}\:{integer}\:{serie}\:{the}\:{functions} \\ $$$${f}\left({x}\right)=\frac{{x}}{{shx}}\:\:\:{and}\:{g}\left({x}\right)=\frac{{x}}{{thx}}\: \\ $$$$\:{Give}\:{the}\:{DL}_{\mathrm{2}} \:{of}\:\:{f}\:{and}\:{g}\:{around}\:{zero}\: \\ $$$${Why}\:{can}'{t}\:{we}\:{use}\:{that}\:{theorem}\:{to}\:{explicit} \\ $$$${f}\left({a}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\:\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} +{a}^{\mathrm{2}} }\:\:\:??? \\ $$

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