Gamma Function Holomorphic
We consider a proof that shows that the Gamma function $\Gamma(z):= \int_0 ^{\infty}t^{z-1}e^{-t}dt$ is holomorphic on the complex halfplane $\{z \in \mathbb{C} \vert Re(z) >0 \}$:
The proof constructs a function sequence $(I_n)_{n \in \mathbb{N}}$ of holomorphic (?) functions converging to $\Gamma$ for every $z$ with $Re(z) >0$. A theorem from complex analysis says that the limit $\lim_n I(z)_n \to \Gamma(z)$ must be also holomorphic.
In the proof the $I_n$ are defined as
$$I_n := \int_{1/n} ^{n}t^{z-1}e^{-t}dt $$
Question: How to show formally that all $I_n$ are holomorphic functions on the $Re( z) >0$-halfplane?
Clear is that in each $z \in \{Re>0 \}$ the integral $I_n(z)$ has a finite value. The problem is the integrand $t^{z-1}$: Why is it holomorphic?
Best Answer
Because $t^{z-1}=e^{(z-1)\log t}$.