Calculate the $r$-th derivatives of $Γ(z,v), r=1,2,...$

Question

Let us consider the the incomplete Γ-function $$Γ(z,v)=∫_{v}^{+∞}t^{z-1}e^{-t}dt$$

My question is: Calculate the $r$-th derivatives of $Γ(z,v), r=1,2,...$ with respect to $z$.

Answer 1

We assume $v\gt 0$. It's tempting to take the derivative under the integral sign, but we have to check the conditions. We define $$f(z,t):=t^{z-1}e^{-t}.$$ Since $\partial_z f(z,t)=(z-1)t^{z-2}e^{-t}+t^{z-1}e^{-t},$ for a fixed $z_0$, there is $a\gt 0$ such that $|\partial_zf(z,t)|\leqslant g(t)$ for $|z-z_0|\lt r$ and $g$ is integrable on $[v,\infty)$ (take $g(t):=4t^2e^{-t}$).

One can iterate the process.