Modulus of Gamma Function of Imaginary Number

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Theorem

Let $t \in \R$ be a real number.

Then:

$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$

where:

$\Gamma$ is the Gamma function
$\csch$ is the hyperbolic cosecant function.


Proof

By Euler's Reflection Formula:

$\map \Gamma {i t} \, \map \Gamma {1 - i t} = \pi \, \map \csc {\pi i t}$

From Gamma Difference Equation:

$-i t \, \map \Gamma {i t} \, \map \Gamma {-i t} = \pi \, \map \csc {\pi i t}$

Then:

\(\ds \cmod {-i t} \cmod {\map \Gamma {i t} } \cmod {\map \Gamma {-i t} }\) \(=\) \(\ds \cmod t \cmod {\map \Gamma {i t} } \cmod {\overline {\map \Gamma {i t} } }\) Complex Conjugate of Gamma Function
\(\ds \) \(=\) \(\ds \cmod t \cmod {\map \Gamma {i t} }^2\) $\cmod z = \cmod {\overline z}$

and:

\(\ds \cmod {\pi \, \map \csc {\pi i t} }\) \(=\) \(\ds \cmod {-i \pi \, \map \csch {\pi t} }\) Hyperbolic Sine in terms of Sine
\(\ds \) \(=\) \(\ds \pi \, \map \csch {\pi \cmod t}\) Hyperbolic Sine Function is Odd

So:

$\cmod {\map \Gamma {i t} }^2 = \dfrac {\pi \, \map \csch {\pi \cmod t} } {\cmod t}$

As $\cmod z \ge 0$ for all complex numbers $z$, we can take the non-negative square root and write:

$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \, \map \csch {\pi \cmod t} } {\cmod t} }$

However, by Hyperbolic Sine Function is Odd:

$\dfrac {\pi \, \map \csch {-\pi t} } {-t} = \dfrac {-\pi \, \map \csch {\pi t} } {-t} = \dfrac {\pi \, \map \csch {\pi t} } t$

Hence we can remove the modulus and simply write:

$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$

$\blacksquare$


Also reported as

This result can also be seen reported as:

$\cmod {\map \Gamma {i t} }^2 = \dfrac \pi {t \sinh \pi t}$


Sources