Dear all, I am looking for explicit (at least more explicit than the original expression) for

1) Re$(\Gamma(a, i\omega))$

as well as

2) Im$(\Gamma(a, i\omega)),$

where i Re and Im denote the real and imaginary part, and $\Gamma(a, i\omega)$ is the Incomplete Gamma function with the arguments $a$ and $i\omega$. The letter $i$ denotes imaginary unit, $a>0$ is a real number and $\omega$ is also a real number.

I would presume that 1) and 2) could be written as functions of $\Gamma(a, \omega)$, however by application of change of variables in the integral definition of the Incomplete Gamma function did not succeed for me.

Thank you in advance for any kind of input.