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Shows-that-1-ix-2-pi-xsinh-pix-with-z-0-t-z-1-e-t-dt-Then-Prove-that-0-1-ix-2-dx-pi-4-




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)
$$\:{Shows}\:{that}\:\:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} =\frac{\pi}{{xsinh}\left(\pi{x}\right)}\:\:\:\:\:\:{with}\:\Gamma\left({z}\right)=\int_{\mathrm{0}_{} } ^{\infty} \:{t}^{{z}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$${Then}\:{Prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} \:{dx}\:=\frac{\pi}{\mathrm{4}} \\ $$