Gamma function
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In mathematics, the Gamma function extends the factorial function to complex and non-integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). The factorial function of an integer n is written n! and is equal to the product n! = 1 × 2 × 3 × ⋯ × n. The Gamma function "fills in" the factorial function for non-integer and complex values of n. If z is a real variable, then for natural number values only, we have
- [\Gamma(z+1)=z!\, ]
Because the gamma and factorial functions grow so rapidly for moderately-large arguments, many computing environments include an ln(gamma) function that returns the natural logarithm of the gamma function: this grows much more slowly, and for combinatorial calculations allows adding and subtracting logs instead of multiplying and dividing very large values.
Definition
The notation Γ(z) is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral
- [\Gamma(z) = \int_0^\infty t^\,e^\,dt]
- [\Gamma(z+1)=z \, \Gamma(z)\,.]
- [\Gamma(n+1) = n \, \Gamma(n) = \cdots = n! \, \Gamma(1) = n!\,]
It is a meromorphic function of x with simple poles at x = -n (n = 0, 1, 2, 3, ...) and residues (-1)n/n!. George Allen, and Unwin, Ltd., The Universal Encyclopedia of Mathematics. United States of America, New American Library, Simon and Schuster, Inc., 1964. (Forward by James R. Newman) 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.
Alternative definitions
The following infinite product definitions for the Gamma function, due to Euler and Weierstrass respectively, are valid for all complex numbers z which are not negative integers or zero
- [\Gamma(z) = \lim_ \frac]
- [\Gamma(z) = \frac} \prod_^\infty \left(1 + \frac\right)^ e^]
Properties
Other important functional equations for the Gamma function are Euler's reflection formula
- [\Gamma(1-z) \; \Gamma(z) = ]
- [\Gamma(z) \; \Gamma\left(z + \frac\right) = 2^ \; \sqrt \; \Gamma(2z).]
- [\Gamma(z) \; \Gamma\left(z + \frac\right) \; \Gamma\left(z + \frac\right) \cdots\Gamma\left(z + \frac\right) =(2 \pi)^ \; m^ \; \Gamma(mz).]
- [\Gamma\left(\frac\right)=\sqrt,]
The derivatives of the Gamma function are described in terms of the polygamma function. For example:
- [\Gamma'(z)=\Gamma(z)\psi_0(z).\,]
- [\operatorname(\Gamma,-n)=\frac.]
An alternative notation which was originally introduced by Gauss and which is sometimes used is the Pi function, which in terms of the Gamma function is
- [\Pi(z) = \Gamma(z+1) = z \; \Gamma(z),]
- [\Pi(n) = n!.\,]
- [\Pi(z) \; \Pi(-z) = \frac = \frac(z)}]
- [\Pi\left(\frac\right) \, \Pi\left(\frac\right) \cdots \Pi\left(\frac\right)=\left(\frac\right)^ \, m^ \, \Pi(z).]
- [\pi(z) = \,]
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 Gamma function is related to the Beta function by the formula
- [\Beta(x,y)=\frac.]
The analog of the Gamma function over a finite field or a finite ring are the Gaussian sums, a type of exponential sum.
The reciprocal Gamma function is an entire function and has been studied as a specific topic.
Plots
Particular values
Main article: Particular values of the Gamma function
[\Gamma(-3/2)\,] [= \frac } \,] [\Gamma(-1/2)\,] [= -2\sqrt\,] [\Gamma(1/2)\,] [= \sqrt\,] [\Gamma(1)\,] [=0!=1 \,] [\Gamma(3/2)\,] [= \frac } \,] [\Gamma(2)\,] [=1!=1 \,] [\Gamma(5/2)\,] [= \frac } \,] [\Gamma(3)\,] [=2!=2 \,] [\Gamma(7/2)\,] [= \frac } \,] [\Gamma(4)\,] [=3!=6 \,] Approximations
Complex values of the Gamma function can be computed numerically with arbitrary precision using Stirling's approximation or the Lanczos approximation.By partial integration of Euler's integral, the Gamma function can also be written
- [\Gamma(z) = x^z e^ \sum_^\infty \frac + \int_x^\infty e^ t^ dt]
For arguments that are integer multiples of 1/24 the Gamma function can also be evaluated quickly using arithmetic-geometric mean iterations (see particular values of the Gamma function).
See also
- Beta function
- Bohr-Mollerup theorem
- Digamma function
- Elliptic gamma function
- Gamma distribution
- Gauss's constant
- Incomplete gamma function
- Multivariate Gamma function
- Polygamma function
- Stirling's approximation
- Trigamma function
References and further reading
- General
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [(See Chapter 6)]
- , [Gamma function] at MathWorld.
- Pascal Sebah and Xavier Gourdon. Introduction to the Gamma Function. In [PostScript] and [HTML] formats.
- Bruno Haible & Thomas Papanikolaou. [Fast multiprecision evaluation of series of rational numbers]. Technical Report No. TI-7/97, Darmstadt University of Technology, 1997
External links
- Examples of problems involving the Gamma function can be found at [Exampleproblems.com].
- [Gamma function calculator]
- [Wolfram gamma function evaluator (arbitrary precision)]
- [Gamma] at the Wolfram Functions Site.
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