Gamma function
From Wikipedia, the free encyclopedia.
In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. The notation is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral
-
converges absolutely. Using
integration by parts, one can show that
Because of Γ(1) = 1, this relation implies
-
for all
natural numbers n. It can further be used to extend Γ(
z) to a
meromorphic function defined for all complex numbers
z except
z = 0, − 1,− 2, − 3, ... by
analytic continuation.
It is this extended version that is commonly referred to as the Gamma function.
An alternative notation which is somtimes used is the
Pi function, which in terms of the Gamma function is
-
We also sometimes find
-
which is an
entire function, defined for every complex number. That is entire entails it has no poles, so has no zeros.
Perhaps the most well-known value of the Gamma function at a non-integer is
The Gamma function has a
pole of order 1 at
z = −
n for every
natural number n; the
residue there is given by
The following multiplicative form of the Gamma function is valid for all complex numbers
z which are not non-positive integers:
-
Here γ is the
Euler-Mascheroni constant.
Relation to other functions
In the first integral above, which defines the Gamma function, the limits of integration are fixed.
The incomplete gamma function is the function obtained by allowing either the upper or lower limit of integration to be variable.
The derivative of the logarithm of the Gamma function is called the digamma function.
References
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6.)
- G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
- W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 6.1.)
External links
