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
- 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $15$: Miscellaneous Examples
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.17$: Miscellaneous Results