Gamma function
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
Referenced By
Analytic continuation | Bessel differential equation | Bessel function | Beta function | Chi-square | Chi-square distribution | Chi-squared distribution | Digamma function | Elementary function | Elementary functions | Euler-Mascheroni constant | Euler-Mascheroni gamma constant | Euler gamma | Euler integral | Euler mascheroni | Factorial | Factorial function | Incomplete gamma function | Infinite product | Letters used in Maths and Science | List of complex analysis topics | List of functions | List of letters used in mathematics and science | List of mathematical functions | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | List of real analysis topics | Meromorphic | Meromorphic function | Morera's theorem | Polygamma function | Riemann Zeta function | Special function | Special functions | Transcendental number | Trascendental number
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