GlobalCharacterizationOfHypergeometricFunction 2007-10-28 03:53:49 2007-10-28 04:04:58 Definition hypergeometric function % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Riemann noted that the hypergeometric function can be characterized by its global properties, without reference to power series, differential equations, or any other sort of explicit expression. His characterization is conveniently restated in terms of sheaves: Suppose that we have a sheaf of holomorphic functions over $\mathbb{C} \setminus \{0,1\}$ which satisfy the following properties: \begin{itemize} \item It is closed under analytic continuation. \item It is closed under taking linear combinations. \item The space of function elements over any open set is two dimensional. \item If one analytically continues a function element along a line towards one of the points $0,1$ or towards infinity, it is bounded by a power. \end{itemize} Then the sheaf consists of solutions to a hypergeometric equation, hence the function elements are hypergeometric functions.