Statistics How To

Gamma Function: Definition, Properties

Statistics Definitions > Gamma Function

You may want to read this article first: What is a Factorial?

What is a Gamma Function?

The Gamma function (sometimes called the Euler Gamma function) is an advanced mathematical function with some unusual properties. The usual definition for the function is the following improper integral:
gf 1


This definition is related to factorials by the following formula:
Γ(n) = (x – 1)!.
In other words, the gamma function is equal to the factorial function. However, while the factorial function is only defined for non-negative integers, the gamma can handle fractions as well as complex numbers.

For positive values x < 1. the integrand becomes infinitely large as t approaches 0. For values of x greater than or equal to 1, the series is convergent.

In Excel

In Microsoft Excel, the Gamma Function (which uses the above definition) is:
GAMMA(number)
For values of 0 or less, GAMMA will return an error (#NUM!).

Alternate Definition

A second, similar, definition is:
gf3

Notation



The notation Γ(n) = (x – 1)! is due to Legendre. It is the most common form for the gamma function.
Equivalent, rarely used, versions of the formula include:

  • Π (n) = n! (from Gauss) and
  • (z – 1)!.

Particular Values for the Gamma Function

y = Γ(x) : The Gamma Function.

y = Γ(x) : The Gamma Function.




Particular values include:

  • Γ(-1) = (-1 – 1)! = -2! = ∞.
  • Γ(0) = (0 – 1)! = -1! = ∞.
  • Γ(1/2) = (1/2 – 1)! = -1/2! = √π.
  • Γ(1) = (1 – 1)! = 0! = 1.
  • Γ(4) = (4 – 1)! = 3 x 2 x 1 = 6.

Any whole negative number leads to a value of infinity, while any positive whole number leads to a value of 1. This is one of the strange results the formula is famous for. The function is also well known in probability for the result Γ(1/2) = √π.

Next: The Multivariate Gamma Function.

References:
Bateman, H & Erdélyi, A. (1955) “Higher Transcendental Functions” [in 3 volumes]. Mc Graw-Hill Book Company.
Davis, P. J. (1959). “Leonhard Euler’s Integral”, The American Mathematical Monthly, Vol. 66, No. 10 (Dec., 1959), pp. 849–869.
Note: The Mathematical Association of America has a scan of Euler’s original 1938 work on the function. You can find it here.

------------------------------------------------------------------------------

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!
Gamma Function: Definition, Properties was last modified: October 12th, 2017 by Stephanie Glen