Generalized series expansion for $\Gamma(a,z)$ in $a$ at $a=0$

Question

I need the generalized series expansion of $\Gamma(a,z)$ for $z\in\mathbb{C}$ and $a\to0^+$. Mathematica gives a result that seems to be correct, but I have to make sure of its validity.

I came across this expansion on Function Wolfram Research, but it does not seem to work for the case that I am interested in, namely $a_0 = 0$. It might be possible to take the limit of the seemingly divergent terms (e.g., $\frac{1}{a_0^2}$), and I think that's how Mathematica computes the expansion.

I also found an expansion on DLMF for $\Gamma(a,x)$ where $x\in \mathbb{R}$: 8.7.6. Although not stated, it seems that it holds only for $a>0$.

The leading term of the expansion of $\Gamma(a,z)$ in $a$ at $a=0$, if exists, should be $\Gamma(0,z)$ which contains a $\log z$ term.

Any thoughts?

Answer 1

You can have this expansion

$$ \Gamma(a,z)={\rm Ei} \left( 1,z \right) + b(z)a + O( a^{2}), $$

where

$$ b(z) = {\rm Ei} ( 1,z) \ln \left( z \right) + \frac{{\gamma}^{2}}{2}+\ln( z )\gamma + \frac{1}{2} ( \ln( z))^{2}+\frac{{\pi }^{2}}{12}-z\,{\mbox{$_3$F$_3$}(1,1,1;\,2,2,2;\,-z)},$$

where $ {\mbox{$_3$F$_3$}(1,1,1;\,2,2,2;\,-z)} $ is the generalized hypergeometric function.