gamma function
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Introduction
The gamma function can be thought of as the natural way to generalize the concept of the factorial to non-integer arguments.
Leonhard Euler (http://planetmath.org/EulerLeonhard) came up with a formula for such a generalization in 1729. At around the same time, James Stirling independently arrived at a different formula, but was unable to show that it always converged. In 1900, Charles Hermite showed that the formula given by Stirling does work, and that it defines the same function as .
Definitions
However, it is now more commonly defined by
for with , and by analytic continuation for the rest of the complex plane, except for the non-positive integers (where it has simple poles).
Functional equations
The gamma function satisfies the functional equation
except when is a non-positive integer. As , it follows by induction that
for positive integer values of .
Another functional equation satisfied by the gamma function is
for non-integer values of .
Approximate values
The gamma function for real looks like this:
(generated by GNU Octave and gnuplot) |
It can be shown that . Approximate values of for some other are:
If the value of is known for some , then one may calculate the value of for any integer by making use of the formula . We have
which is easy to calculate if we know .
References
- 1 Julian Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003. (Chapter 6 is about the gamma function.)
Title | gamma function |
---|---|
Canonical name | GammaFunction |
Date of creation | 2013-03-22 12:00:39 |
Last modified on | 2013-03-22 12:00:39 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 44 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 30D30 |
Classification | msc 33B15 |
Synonym | gamma-function |
Synonym | -function |
Synonym | Euler’s gamma function |
Synonym | Euler’s gamma-function |
Synonym | Euler’s -function |
Related topic | BohrMollerupTheorem |
Related topic | TableOfLaplaceTransforms |