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Question Number 67540 by mathmax by abdo last updated on 28/Aug/19

prove that   ∣Γ((1/2)+it)∣ =(√((2π)/(e^(πt)  +e^(−πt) )))  and ∣Γ(1+it)∣ =(√((2πt)/(e^(πt) −e^(−πt) )))

$${prove}\:{that}\:\:\:\mid\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\mid\:=\sqrt{\frac{\mathrm{2}\pi}{{e}^{\pi{t}} \:+{e}^{−\pi{t}} }} \\ $$$${and}\:\mid\Gamma\left(\mathrm{1}+{it}\right)\mid\:=\sqrt{\frac{\mathrm{2}\pi{t}}{{e}^{\pi{t}} −{e}^{−\pi{t}} }} \\ $$

Commented by ~ À ® @ 237 ~ last updated on 28/Aug/19

    I ′m not sure  if we always have Γ^− (z)=Γ(z^− )  But if  is true   ∣Γ((1/2)+it)∣^2 =Γ((1/2)+it)Γ((1/2)−it)=Γ((1/2)+it)Γ(1−((1/2)+it))    =(π/(sin((π/2)+iπt)))     using  complements formulas  Γ(s)Γ(1−s)=(π/(sin(πs)))    =(π/(cos(iπt)))=((2π)/(e^(πt) +e^(−πt) ))   ∣Γ(1+it)∣^2 =Γ(1+it)Γ(1−it)=itΓ(it)Γ(1−it)=((iπt)/(sin(iπt)))=((iπt)/((e^(−πt) −e^(πt) )/(2i)))=((2πt)/(e^(πt) −e^(−πt) ))

$$ \\ $$$$ \\ $$$${I}\:'{m}\:{not}\:{sure}\:\:{if}\:{we}\:{always}\:{have}\:\overset{−} {\Gamma}\left({z}\right)=\Gamma\left(\overset{−} {{z}}\right) \\ $$$${But}\:{if}\:\:{is}\:{true}\: \\ $$$$\mid\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\mid^{\mathrm{2}} =\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}−{it}\right)=\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\Gamma\left(\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\right) \\ $$$$\:\:=\frac{\pi}{{sin}\left(\frac{\pi}{\mathrm{2}}+{i}\pi{t}\right)}\:\:\:\:\:{using}\:\:{complements}\:{formulas}\:\:\Gamma\left({s}\right)\Gamma\left(\mathrm{1}−{s}\right)=\frac{\pi}{{sin}\left(\pi{s}\right)} \\ $$$$\:\:=\frac{\pi}{{cos}\left({i}\pi{t}\right)}=\frac{\mathrm{2}\pi}{{e}^{\pi{t}} +{e}^{−\pi{t}} }\: \\ $$$$\mid\Gamma\left(\mathrm{1}+{it}\right)\mid^{\mathrm{2}} =\Gamma\left(\mathrm{1}+{it}\right)\Gamma\left(\mathrm{1}−{it}\right)={it}\Gamma\left({it}\right)\Gamma\left(\mathrm{1}−{it}\right)=\frac{{i}\pi{t}}{{sin}\left({i}\pi{t}\right)}=\frac{{i}\pi{t}}{\frac{{e}^{−\pi{t}} −{e}^{\pi{t}} }{\mathrm{2}{i}}}=\frac{\mathrm{2}\pi{t}}{{e}^{\pi{t}} −{e}^{−\pi{t}} }\: \\ $$$$ \\ $$

Commented by mathmax by abdo last updated on 29/Aug/19

thank you sir.

$${thank}\:{you}\:{sir}. \\ $$

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