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.


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.

  • $\begingroup$ 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. $\endgroup$ – Petern Jan 26 '13 at 10:55
  • $\begingroup$ 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 $\endgroup$ – Petern Feb 7 '13 at 22:18

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