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Question Number 154235 by mnjuly1970 last updated on 15/Sep/21

prove that: Im( ψ ( i ) )= (( 1)/( 2)) + (( π)/2) coth(π ) m.n

$$ \\ $$$$\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\mathrm{I}{m}\left(\:\psi\:\left(\:{i}\:\right)\:\right)=\:\frac{\:\mathrm{1}}{\:\mathrm{2}}\:+\:\frac{\:\pi}{\mathrm{2}}\:{coth}\left(\pi\:\right) \\ $$$$\:\:\:\:\:\:\:\:{m}.{n} \\ $$

Answered by mindispower last updated on 16/Sep/21

Ψ(1−i)−Ψ(i)=πcot(iπ)....1 using Ψ(1−x)−Ψ(x)=πcot(πx) Ψ(1+(−i))=(1/(−i))+Ψ(−i) ImΨ(−i)=−Im(Ψ(i)) ⇒Im(Ψ(1−i))=1−Im(Ψ(i)) 1⇒−2Im(Ψ(i))+1=Im(πcot(iπ)) ⇒ImΨ(i)=(1/2)+((πcoth(π))/2) ⇒

$$\Psi\left(\mathrm{1}−{i}\right)−\Psi\left({i}\right)=\pi{cot}\left({i}\pi\right)....\mathrm{1} \\ $$$${using}\:\Psi\left(\mathrm{1}−{x}\right)−\Psi\left({x}\right)=\pi{cot}\left(\pi{x}\right) \\ $$$$\Psi\left(\mathrm{1}+\left(−{i}\right)\right)=\frac{\mathrm{1}}{−{i}}+\Psi\left(−{i}\right) \\ $$$${Im}\Psi\left(−{i}\right)=−{Im}\left(\Psi\left({i}\right)\right) \\ $$$$\Rightarrow{Im}\left(\Psi\left(\mathrm{1}−{i}\right)\right)=\mathrm{1}−{Im}\left(\Psi\left({i}\right)\right) \\ $$$$\mathrm{1}\Rightarrow−\mathrm{2}{Im}\left(\Psi\left({i}\right)\right)+\mathrm{1}={Im}\left(\pi{cot}\left({i}\pi\right)\right) \\ $$$$\Rightarrow{Im}\Psi\left({i}\right)=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\pi{coth}\left(\pi\right)}{\mathrm{2}} \\ $$$$\Rightarrow \\ $$

Commented by mnjuly1970 last updated on 16/Sep/21

thanks alot...mr power

$$\:{thanks}\:{alot}...{mr}\:{power} \\ $$

Commented by mindispower last updated on 16/Sep/21

pleasur

$${pleasur} \\ $$

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