## The gamma function

### Proving holomorphicity and other properties

The gamma funcion $\Gamma$ is a special function in mathematical analysis. For complex $z$ in the right half plane, we define $$\Gamma(z) := \int_{0}^{\infty}t^{z-1}e^{-t}dt. \label{eq1}$$ It turns out that $\Gamma$ is holomorphic and satisfies $\Gamma(z + 1) = z\Gamma(z)$ for all $z$ with Re $z > 0$. This last property means that the gamma function can be thought of as a generalization of the factorial function to real and complex numbers. To be more precise, if $n$ is an integer then $\Gamma(n) = (n-1)!$.

## A Borel set that is neither $F_{\sigma}$ nor $G_{\delta}$
$F_{\sigma}$ and $G_{\delta}$ sets are two classes of sets which are quite ubiquitous in the Borel $\sigma$-algebra of $\mathbb{R}^{n}$. Their names come from a combination of French and German. $F_{\sigma}$ comes from the French words “fermé” meaning “closed” and “somme” meaning, well, “sum”. Together the meaning is a union of closed sets. So a set is $F_{\sigma}$ if it is a countable union of closed sets. Similarly, $G_{\delta}$ comes from the German words “gebiet” meaning an open set and “durchschnitt” meaning intersection. So a set is $G_{\delta}$ if it the countable intersection of open sets.