# Gamma function

The gamma function along part of the real axis
In mathematics, the **gamma function** (Γ(*z*)) is a key topic in the field of special functions. Γ(*z*) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as WIKINIUM\_MATH\_PH\_0

The gamma function is defined for all complex numbers, but it is not defined for the non-positive integers. For a complex number whose real part is a positive integer, the function is defined by:

 WIKINIUM\_MATH\_PH\_2
## Properties

### Proof of factorial representation

The factorial definition of the gamma function can be derived using integration by parts. WIKINIUM\_MATH\_PH\_4 Let WIKINIUM\_MATH\_PH\_6 so WIKINIUM\_MATH\_PH\_8 and WIKINIUM\_MATH\_PH\_10 so WIKINIUM\_MATH\_PH\_12 WIKINIUM\_MATH\_PH\_14 so by applying integration by parts,
 WIKINIUM\_MATH\_PH\_16
as the first part vanishes (equals 0). Note that WIKINIUM\_MATH\_PH\_18 .
Thus, WIKINIUM\_MATH\_PH\_20 for WIKINIUM\_MATH\_PH\_22 , that is, real part of WIKINIUM\_MATH\_PH\_24 is positive.

### Particular values

Some particular values of the gamma function are:

 WIKINIUM\_MATH\_PH\_26
### Pi function

Gauss introduced the **Pi function**. This is another way of denoting the gamma function. In terms of the gamma function, the Pi function is

 WIKINIUM\_MATH\_PH\_28
so that

 WIKINIUM\_MATH\_PH\_30
for every non-negative integer *n*.

## Applications

### Analytic number theory

The gamma function is used to study the Riemann zeta function. A property of the Riemann zeta function is its functional equation:

 WIKINIUM\_MATH\_PH\_32
Bernhard Riemann found a relation between these two functions. This was published in his 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Prime Numbers less than a Given Quantity")

 WIKINIUM\_MATH\_PH\_34
## Related pages

* Gamma distribution

* Milton Abramowitz and Irene A. Stegun, eds. *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.* New York: Dover, 1972. *(See Chapter 6)*
* Andrews, George E.; Askey, Richard; Roy, Ranjan (2000). *Special Functions*. Cambridge University Press. ISBN 978-0-521-78988-2.
* Emil Artin, "The Gamma Function", in Rosen, Michael (ed.) *Exposition by Emil Artin: a selection*; History of Mathematics 30. Providence, RI: American Mathematical Society (2006).
* Birkhoff, George D. (1913). "Note on the gamma function". *Bull. Amer. Math. Soc*. **20** (1): 1–10. doi:10.1090/S0002-9904-1913-02429-7. MR 1559418. S2CID 120440543.
* P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
* Bonnar, James (2010). *The Gamma Function*. Createspace Independent Publishing. ISBN 978-1-4636-9429-6.
* Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," *American Mathematical Monthly* **66**, 849-869 (1959)
* Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.1. Gamma Function", *Numerical Recipes: The Art of Scientific Computing* (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the original on 2021-10-27, retrieved 2013-02-10
* O. R. Rocktaeschel, ´´Methoden zur Berechnung der Gammafunktion für komplexes Argument``, University of Dresden, Dresden, 1922.
* Temme, Nico M. (1996). *Special Functions: An Introduction to the Classical Functions of Mathematical Physics*. John Wiley & Sons. ISBN 978-0-471-11313-3.
* Whittaker, E.T.; Watson, G.N. (1996). *A Course of Modern Analysis*. Cambridge University Press. ISBN 978-0-521-58807-2.