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 .
However, it is now more commonly defined by
Another equivalent definition is
where is Euler’s constant.
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 .
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 .
- 1 Julian Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003. (Chapter 6 is about the gamma function.)
|Date of creation||2013-03-22 12:00:39|
|Last modified on||2013-03-22 12:00:39|
|Last modified by||yark (2760)|
|Synonym||Euler’s gamma function|