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Question Number 65786 by ~ À ® @ 237 ~ last updated on 04/Aug/19
Shows that ∣Γ(1+ix)∣^2 =(π/(xsinh(πx))) with Γ(z)=∫_0_ ^∞ t^(z−1) e^(−t) dt Then Prove that ∫_0 ^∞ ∣Γ(1+ix)∣^2 dx =(π/4)
$$\:{Shows}\:{that}\:\:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} =\frac{\pi}{{xsinh}\left(\pi{x}\right)}\:\:\:\:\:\:{with}\:\Gamma\left({z}\right)=\int_{\mathrm{0}_{} } ^{\infty} \:{t}^{{z}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$${Then}\:{Prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} \:{dx}\:=\frac{\pi}{\mathrm{4}} \\ $$

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