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 noninteger and complex values of n. If z is a complex variable, then for integer values only, we have

<math>\Gamma(z+1)=z!\, </math>

but for noninteger and complex values of z, the above equation does not apply, since the factorial function is not defined.

Because the gamma and factorial functions grow so rapidly for moderately-large arguments, many computing environments include an lngamma 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.

1 Definition
2 Alternative definitions
3 Properties
4 Relation to other functions
5 Plots
6 Particular values
7 Approximations
8 See also
9 References and further readings
10 External links

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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

<math>

\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt </math> converges absolutely. Using integration by parts, one can show that

<math>\Gamma(z+1)=z \, \Gamma(z)\,.</math>

Because Γ(1) = 1, this relation implies that

<math>\Gamma(n+1) = n \, \Gamma(n) = \cdots = n! \, \Gamma(1) = n!\,</math>

for all natural numbers n.

Image:Gamma abs.png It is a meromorphic function of x with simple poles at x = -n (n = 0, 1, 2, 3, ...) and residues (-1)ⁿ/n!. <ref>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)</ref> 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.

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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 non-positive integers:

<math>

\Gamma(z) = \lim_{n \to \infty} \frac{n! \; n^z}{z \; (z+1)\cdots(z+n)} </math>

<math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}</math>

where γ is the Euler-Mascheroni constant.

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Properties

Other important functional equations for the Gamma function are Euler's reflection formula

<math>

\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin \pi z} </math>

and the duplication formula

<math>

\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z). </math>

The duplication formula is a special case of the multiplication theorem

<math>

\Gamma(z) \; \Gamma\left(z + \frac{1}{m}\right) \; \Gamma\left(z + \frac{2}{m}\right) \cdots \Gamma\left(z + \frac{m-1}{m}\right) = (2 \pi)^{(m-1)/2} \; m^{1/2 - mz} \; \Gamma(mz). </math>

Perhaps the most well-known value of the Gamma function at a non-integer argument is

<math>\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi},</math>

which can be found by setting z=1/2 in the reflection formula or by noticing the beta function for (1/2, 1/2), which is <math>\sqrt \pi</math>.

The derivatives of the Gamma function are described in terms of the polygamma function. For example:

<math>\Gamma'(z)=\Gamma(z)\psi_0(z).\,</math>

The Gamma function has a pole of order 1 at z = −n for every natural number n; the residue there is given by

<math>\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}.</math>

The Bohr-Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the Gamma function is log-convex, that is, its natural logarithm is convex.

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

<math>\Pi(z) = \Gamma(z+1) = z \; \Gamma(z),</math>

so that

Using the Pi function the reflection formula takes on the form

<math>\Pi(z) \; \Pi(-z) = \frac{\pi z}{\sin \pi z}

= \frac{1}{\mathrm{sinc}_N(x)}</math>

where sinc_N is the normalized Sinc function, while the multiplication theorem takes on the form

<math>

\Pi\left(\frac{z}{m}\right) \, \Pi\left(\frac{z-1}{m}\right) \cdots \Pi\left(\frac{z-m+1}{m}\right) = \left(\frac{(2 \pi)^m}{2 \pi m}\right)^{1/2} \, m^{-z} \, \Pi(z). </math>

We also sometimes find

which is an entire function, defined for every complex number. That π(z) is entire entails it has no poles, so Γ(z) has no zeros.

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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

<math>

\Beta(x,y)=\frac{\Gamma(x) \; \Gamma(y)}{\Gamma(x+y)}. </math>

The derivative of the logarithm of the Gamma function is called the digamma function; higher derivatives are the polygamma functions.

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.

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Plots

Real part of Γ(z)

Imaginary part of Γ(z)

Absolute value of Γ(z)

Real part of log Γ(z)

Imaginary part of log Γ(z)

Absolute value of log Γ(z)

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Particular values

Main article: Particular values of the Gamma function

<math>\Gamma(-2)\,</math>	(undefined)
<math>\Gamma(-3/2)\,</math>	<math>= \frac {4\sqrt{\pi}} {3} \,</math>
<math>\Gamma(-1)\,</math>	(undefined)
<math>\Gamma(-1/2)\,</math>	<math>= -2\sqrt{\pi}\,</math>
<math>\Gamma(0)\,</math>	(undefined)
<math>\Gamma(1/2)\,</math>	<math>= \sqrt{\pi}\,</math>
<math>\Gamma(1)\,</math>	<math>=0!=1 \,</math>
<math>\Gamma(3/2)\,</math>	<math>= \frac {\sqrt{\pi}} {2} \,</math>
<math>\Gamma(2)\,</math>	<math>=1!=1 \,</math>
<math>\Gamma(5/2)\,</math>	<math>= \frac {3 \sqrt{\pi}} {4} \,</math>
<math>\Gamma(3)\,</math>	<math>=2!=2 \,</math>
<math>\Gamma(7/2)\,</math>	<math>= \frac {15\sqrt{\pi}} {8} \,</math>
<math>\Gamma(4)\,</math>	<math>=3!=6 \,</math>

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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

<math>\Gamma(z) = x^z e^{-x} \sum_{n=0}^\infty \frac{x^n}{z(z+1) \dots (z+n)} + \int_x^\infty e^{-t} t^{z-1} dt</math>

where, if Re(z) has been reduced to the interval [1, 2], the last integral is smaller than <math>xe^{-x} < 2^{-N}</math>. Thus by choosing an appropriate x, the Gamma function can be evaluated to N bits of precision with the above series. If z is rational, the computation can be performed with binary splitting in time <math>O((\log N)^2 M(N))</math> where M(N) is the time needed to multiply two N-bit numbers.

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).

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References and further readings

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)
G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
Harry Hochstadt. The Functions of Mathematical Physics. New York: Dover, 1986 (See Chapter 3.)
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.)