Show that the gamma function
$$\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$$
Is holomorphic in $\{Re(z)>0\}$.
Ok, so I was told to try to solve this excercise by defining a sequence of functions
$$f_n=\int_{\frac{1}{n}}^{n}e^{-t}t^{z-1}dt$$
And try to bound the $n$th term by a converging integral (I guess) in any compact subset $K \subset \{Re(z)>0\}$. However, Im not sure what is that I have to bound, and how. Any hints?
Best Answer
I suggest using Morera's Theorem instead, which states that a continuous function $f$ is holomorphic in an open region $\Omega$ if the integral of $f$ over every triangle $\Delta$ is $0$ whenever the triangle and its interior is contained in $\Omega$.
The integral function you have for $\Gamma(z)$ is continuous by the Lebesgue dominated convergence theorem. You can use Fubini's theorem to interchange orders of integration in order to prove that $$ \oint_{\Delta} \Gamma(z)dz = \int_{0}^{\infty}e^{-t}\left(\oint_{\Delta}e^{(-1+z)\ln t}dz\right)dt = 0. $$