# Imaginary parts of GAMMA(1/2+I*y)

1. Oct 14, 2009

### Mathjunkie

Hi:

Does anyone know of an explicit formula for the Real and Imaginary parts of GAMMA(1/2+I*y) as functions of y ?

|GAMMA(1/2+I*y)|^2 =Re(GAMMA(1/2+I*y))^2+Im(GAMMA(1/2+I*y))^2= Pi/cosh(Pi*y)

but can't find anything about each of the Real and Imaginary terms individually. Checked just about every reference book that exists, and tried to derive something myself with no luck.

Thanks to anyone who can point me to a reference. You'd think that something like that must be known.

2. Oct 15, 2009

### g_edgar

Re: GAMMA(1/2+I*y)

Perhaps not what you wanted...
Real part
$$\int _{0}^{\infty }\!{{\rm e}^{-t}}\sqrt {t}\cos \left( y\ln \left( t \right) \right) {dt}$$
Imaginary part
$$\int _{0}^{\infty }\!{{\rm e}^{-t}}\sqrt {t}\sin \left( y\ln \left( t \right) \right) {dt}$$