## Welcome to the The Gamma Function Wiki

The Gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. Although the Gamma function is defined for all complex numbers except the non-positive integers, it is defined via an improper integral that converges only for complex numbers with a positive real part.

This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we know as the Gamma function.

The Gamma function has important applications in probability theory, combinatorics and most, if not all, areas of physics.

## Definition

## Important Topics Related to the Gamma Function

- the analytic continuation of the factorials
- properties of the gamma function via complex analysis
- convexity and the gamma function
- the Beta Function
- Wallis's Integrals
- Wallis's Product
- Legendre's Duplication Formula
- Gauss's Multiplication Formula
- Weierstrass' Product Form of the Gamma Function
- Euler's Reflection Formula
- Half-integer Values
- Digamma and Polygamma Functions
- Series Expansions
- Euler-Mascheroni Integrals
- Relationships between the Gamma and zeta Functions
- Stirling's Formula
- Weierstrass Factor Theorem
- Mittag-Leffler Theorem

## References

- J. Bonnar, "The Gamma Function", CreateSpace, Seattle, WA, ISBN 1463694296, 2010.
- G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press; 1st Edition, 2001.
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, Dover Publications; 1965.
- N.N. Lebedev, Special Functions and Their Applications, Dover Publications; Revised Edition, 1972.
- W. Rudin, Real and Complex Analysis, McGraw-Hill; 3rd Edition, 1986.
- I. Gradshteyn and I. Ryzik, Table of Integrals, Series and Products, Academic Press; 2nd Edition, 1980 .
- W. Bell, Special Functions for Scientists and Engineers, Dover Publications; 2004.