# The real and imaginary parts of the Incomplete Gamma function for second argument being purely imaginary

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.

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The usual integral representation gives you $$\Gamma(a,i\omega) = \Gamma(a) - (i\omega)^a \int_0^1 t^{a-1} e^{-i\omega t}\ dt$$

I assume you have no trouble writing $(i\omega)^a$ in terms of its real and imaginary parts. As for the integral, Maple says

\eqalign{\int_0^1 t^{a-1} \cos(\omega t)\ dt &= {\frac { \left( {w}^{-a-1/2}\sin \left( w \right) -{w}^{-a+1/2}\cos \left( w \right) \right) {S_1} \left( a+1/2,1/2,w \right) }{a}}\cr &-{\frac {{w}^{-a+1/2}{S_1} \left( a+3/2,3/2,w \right) \sin \left( w \right) }{ \left( 2+a \right) a}}\cr &+{\frac { \left( 2 +a \right) \cos \left( w \right) +w\sin \left( w \right) }{ \left( 2+a \right) a}} \cr}

\eqalign{\int_0^1 t^{a-1} \sin(\omega t)\ dt &=-{\frac {{w}^{-a+1/2}\sin \left( w \right) {S_1} \left( a+1/2,3 /2,w \right) }{ a+1 }}\cr+&{\frac { \left( {w}^{-a +1/2}\cos \left( w \right) -{w}^{-a-1/2}\sin \left( w \right) \right) {S_1} \left( a+3/2,1/2,w \right) }{ \left( a+1 \right) a} }\cr+&{\frac { \left( a+1 \right) \sin \left( w \right) -\cos \left( w \right) w}{ \left( a+1 \right) a}} \cr}

where $S_1$ is the Lommel S1 function.

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Thanks a lot Robert for the hint. I will explore this further after the week-end. I even suspect that the Lommel S1 function might possibly be written in terms of the Incomplete Gamma function, but this I need to look up in detail. Again thank you. – Petern Jan 26 '13 at 10:55
Robert, your hint pointed me towards what I was looking for: the "generalized sine and cosine functions." More infor can be retreived e.g. from here: dlmf.nist.gov/8.21 – Petern Feb 7 '13 at 22:18