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Question Number 117979 by mnjuly1970 last updated on 14/Oct/20

$$...nicecalculus...$$$$provethat:$$$$\mid \mathrm{\Gamma }\left(i\right)\mid \stackrel{?}{=}\sqrt{\frac{\pi }{sinh\left(\pi \right)}}$$$$\mathrm{\Gamma }:\mathcal{E}ulergammafunction$$$$$$$$...m.n.july.1970...$$

Commented by Lordose last updated on 14/Oct/20

THEY ARE NOT EQUAL

Answered by mindispower last updated on 14/Oct/20

$$\mathrm{\Gamma }(1-i)\mathrm{\Gamma }\left(i\right)=\frac{\pi }{sin\left(i\pi \right)}$$$$\mathrm{\Gamma }(1-i)=-i\mathrm{\Gamma }((-i)$$$$\Rightarrow \mathrm{\Gamma }(-i)\mathrm{\Gamma }\left(i\right)=\frac{\pi }{-isin\left(i\pi \right)}=\frac{\pi }{sh\left(\pi \right)}$$$$\mathrm{\Gamma }(-i).\mathrm{\Gamma }\left(i\right)=\frac{\pi }{sh\left(\pi \right)}\Rightarrow \mid \mathrm{\Gamma }\left(i\right){\mid }^2=\frac{\pi }{sh\left(\pi \right)}$$$$\mid \mathrm{\Gamma }\left(i\right)\mid =\sqrt{\frac{\pi }{sh\left(\pi \right)}}$$$$$$

Commented by mnjuly1970 last updated on 14/Oct/20

$$thankyou$$

Commented by mindispower last updated on 14/Oct/20

$$withepleasur$$

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