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

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

Answered by mindispower last updated on 16/Sep/21

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

Commented by mnjuly1970 last updated on 16/Sep/21

$$thanksalot...mrpower$$

Commented by mindispower last updated on 16/Sep/21

$$pleasur$$

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