prove an identity of gamma function

Question

During the process of computing Hawking radiation, I get an expression of gamma function. $$\Gamma( x i) \Gamma(- x i)$$

where x is a real number. Due to some physics motivation, I guess an identity:

$$\Gamma(x i)\Gamma(-i x)=\frac{\pi}{sinh(x)x}$$

And I verified the identity to be true using mathematica( at least for x to be real).

Can some one give me a proof? I am not good at special function. Thanks.

Answer 1

By the reflection formula we have $$\Gamma(1-ix)\Gamma(ix) = \frac{\pi}{\sin{ix}}.$$ Now $\Gamma(1-ix) = -ix\Gamma(-ix)$ (one of the defining properties of $\Gamma$), and we know that $\sinh{x}=-i\sin{(ix)}$. Your result follows.