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 \begin{equation} \Gamma(z) := \int_{0}^{\infty}t^{z-1}e^{-t}dt. \label{eq1} \end{equation} 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 $\Gamma(n) = (n-1)!$.

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The ultimate tmux config

After using Tmux for a year and a half I finally have a configuration that I am satisfied with. I can’t claim originality for a lot of the details here but I try to provide credit where it is due.

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Tags: coding tmux

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.

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