Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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

$${pleasur} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com