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Question Number 133075 by LUFFY last updated on 18/Feb/21

Σ_(n=1) ^∞ ((1/(n^2 +k^2 ))) k∈??????

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{k}^{\mathrm{2}} }\right)\:{k}\in?????? \\ $$

Answered by Dwaipayan Shikari last updated on 18/Feb/21

(1/(2ik))Σ_(n=1) ^∞ (1/(n−ik))−(1/(n+ik))  =(1/(2ik))ψ(ik)−ψ(−ik)=(1/(2ik))(ψ(ik)−ψ(1−ik)+(1/(ik)))  =−(π/(2ik))cot(πik)−(1/(2k^2 ))=−((πi)/(2ik)).((e^(−πk) +e^(πk) )/(e^(−πk) −e^(πk) ))−(1/(2k^2 ))  =(π/(2k)).((e^(πk) +e^(−πk) )/(e^(πk) −e^(−πk) ))−(1/(2k^2 ))=(π/(2k))coth(πk)−(1/(2k^2 ))

$$\frac{\mathrm{1}}{\mathrm{2}{ik}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}−{ik}}−\frac{\mathrm{1}}{{n}+{ik}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{ik}}\psi\left({ik}\right)−\psi\left(−{ik}\right)=\frac{\mathrm{1}}{\mathrm{2}{ik}}\left(\psi\left({ik}\right)−\psi\left(\mathrm{1}−{ik}\right)+\frac{\mathrm{1}}{{ik}}\right) \\ $$$$=−\frac{\pi}{\mathrm{2}{ik}}{cot}\left(\pi{ik}\right)−\frac{\mathrm{1}}{\mathrm{2}{k}^{\mathrm{2}} }=−\frac{\pi{i}}{\mathrm{2}{ik}}.\frac{{e}^{−\pi{k}} +{e}^{\pi{k}} }{{e}^{−\pi{k}} −{e}^{\pi{k}} }−\frac{\mathrm{1}}{\mathrm{2}{k}^{\mathrm{2}} } \\ $$$$=\frac{\pi}{\mathrm{2}{k}}.\frac{{e}^{\pi{k}} +{e}^{−\pi{k}} }{{e}^{\pi{k}} −{e}^{−\pi{k}} }−\frac{\mathrm{1}}{\mathrm{2}{k}^{\mathrm{2}} }=\frac{\pi}{\mathrm{2}{k}}{coth}\left(\pi{k}\right)−\frac{\mathrm{1}}{\mathrm{2}{k}^{\mathrm{2}} } \\ $$

Commented by LUFFY last updated on 19/Feb/21

k belongs to ???????

$${k}\:{belongs}\:{to}\:??????? \\ $$

Commented by Dwaipayan Shikari last updated on 19/Feb/21

k belongs to any number except k=0

$${k}\:{belongs}\:{to}\:{any}\:{number}\:{except}\:{k}=\mathrm{0} \\ $$

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