Lower incomplete gamma

The lower incomplete gamma function is defined for $\mathrm{Re}(a)>0$ by $$\gamma(a,x)=\displaystyle\int_0^x e^{-t}t^{a-1}dt.$$ A single-valued analytic function of $a$ and $x$ can be defined as $$\gamma^*(a,x)=\dfrac{x^{-a}}{\Gamma(a)} \gamma(a,x).$$