Gamma function

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The Gamma function along part of the real axis

In mathematics, the Gamma function is an extension of the factorial function to complex numbers. The Gamma function "fills in" the factorial function for non-integer and complex values of n. For a complex number z with positive real part it is defined by

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,\mathrm{d}t$

which can be extended to the rest of the complex plane, excepting the non-positive integers. If z is a positive integer, then

$\Gamma(z)=(z-1)!\,$

showing the connection to the factorial function.

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

[edit] Definition

The notation Γ(z) is due to Adrien-Marie Legendre. If the real part of the complex number z is positive (Re[z] > 0), then the integral

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,\mathrm{d}t \,\!$

converges absolutely. Using integration by parts, one can show that

$\Gamma(z+1)=z \, \Gamma(z)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1) \,\!$ .

This functional equation generalizes relation n! = n×(n-1)! of the factorial function. We can evaluate Γ(1) analytically:

$\Gamma(1) = \int_0^\infty e^{-t} dt = \lim_{k \rightarrow \infty} -e^{-t} |_0^k = -0 - (-1) = 1$ .

Combining these two relations shows how the factorial function is a special case of the Gamma function:

$\Gamma(n+1) = n \, \Gamma(n) = \cdots = n! \, \Gamma(1) = n!\,$

for all natural numbers n.

The absolute value of the Gamma function on the complex number plane.

It is a meromorphic function of x with simple poles at x = −n (n = 0, 1, 2, 3, ...) and residues (−1)ⁿ/n!. ^[1] 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.

[edit] 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

$\Gamma(z) = \lim_{n \to \infty} \frac{n! \; n^z}{z \; (z+1)\cdots(z+n)} \,\!$

$\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n} \,\!$

where γ is the Euler-Mascheroni constant.

It is straightforward to show that the Euler definition satisfies the functional equation (1) above. Provided z is not equal to 0, -1, -2, ...

$\Gamma(z+1) = \lim_{n \to \infty} \frac{n! \; n^{z+1}}{(z+1) \; (z+2)\cdots(z+1+n)} = \lim_{n \to \infty} \left( z \; \frac{n! \; n^z}{z \; (z+1) \; (z+2)\cdots(z+n)} \; \frac{n}{(z+1+n)}\right) \,\!$

$= z \; \Gamma(z) \; \lim_{n \to \infty} \frac{n}{(z+1+n)} \,\!$

$= z \; \Gamma(z) \,\!$

[edit] Properties

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

$\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin{(\pi z)}} \,\!$

and the duplication formula

$\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z). \,\!$

The duplication formula is a special case of the multiplication theorem

$\Gamma(z) \; \Gamma\left(z + \frac{1}{m}\right) \; \Gamma\left(z + \frac{2}{m}\right) \cdots \Gamma\left(z + \frac{m-1}{m}\right) = (2 \pi)^{(m-1)/2} \; m^{1/2 - mz} \; \Gamma(mz). \,\!$

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

$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}, \,\!$

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 $π$ . In general, for odd integer values of n we have:

$\Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi}\, \frac{n!!}{2^{(n+1)/2}}$ (n odd)

where n!! denotes the double factorial.

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

$\Gamma'(z)=\Gamma(z)\psi_0(z). \,\!$

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

$\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}. \,\!$

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

$\Pi(z) = \Gamma(z+1) = z \; \Gamma(z), \,\!$

so that

$\Pi(n) = n!\, \,\!$

Using the Pi function the reflection formula takes on the form

$\Pi(z) \; \Pi(-z) = \frac{\pi z}{\sin( \pi z)} = \frac{1}{\operatorname{sinc}(z)} \,\!$

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

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

We also sometimes find

$\pi(z) = {1 \over \Pi(z)} \,\!$

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.

[edit] 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

$\Beta(x,y)=\frac{\Gamma(x) \; \Gamma(y)}{\Gamma(x+y)}. \,\!$

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.

The Gamma function also shows up in an important relation with the Riemann zeta function, ζ(z).

$\pi^{-z/2} \; \Gamma\left(\frac{z}{2}\right) \zeta(z) = \pi^{-\frac{1-z}{2}} \; \Gamma\left(\frac{1-z}{2}\right) \; \zeta(1-z) \,\!$

And also in the following elegant formula:

$\zeta(z) \; \Gamma(z) = \int_{0}^{\infty} \frac{u^{z-1}}{e^u - 1} \; \mathrm{d}u \,\!$

[edit] 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)

[edit] Particular values

Main article: Particular values of the Gamma function

$\Gamma(-3/2)\,$	$= \frac {4\sqrt{\pi}} {3} \approx 2.363\,$
$\Gamma(-1/2)\,$	$= -2\sqrt{\pi} \approx -3.545\,$
$\Gamma(1/2)\,$	$= \sqrt{\pi} \approx 1.772\,$
$\Gamma(1)\,$	$=0!=1 \,$
$\Gamma(3/2)\,$	$= \frac {\sqrt{\pi}} {2} \approx 0.886\,$
$\Gamma(2)\,$	$=1!=1 \,$
$\Gamma(5/2)\,$	$= \frac {3 \sqrt{\pi}} {4} \approx 1.329\,$
$\Gamma(3)\,$	$=2!=2 \,$
$\Gamma(7/2)\,$	$= \frac {15\sqrt{\pi}} {8} \approx 3.323\,$
$\Gamma(4)\,$	$=3!=6 \,$

[edit] 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

$\Gamma(z) = x^z e^{-x} \sum_{n=0}^\infty \frac{x^n}{z(z+1) \dots (z+n)} + \int_x^\infty e^{-t} t^{z-1} \mathrm{d}t \,\!$

where, if Re(z) has been reduced to the interval [1, 2], the last integral is smaller than x exp(-x) < 2^-N. 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 O( (log(N)² M(N) ) 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).

Because the Gamma and factorial functions grow so rapidly for moderately-large arguments, many computing environments include a 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. The digamma function, which is the derivative of this function, is also commonly seen.

[edit] Applications

The gamma function is a component in various probability distribution functions, and as such it is applicable in the fields of probability and statistics.

[edit] See also

[edit] 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)
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.)

Citations

Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," Am. Math. Monthly 66, 849-869 (1959)

^ 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)

Eric W. Weisstein, 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

[edit] External links

Cephes - C and C++ language special functions math library
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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Categories: Gamma and related functions | Special hypergeometric functions

Gamma function

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Alternative definitions

[edit] Properties

[edit] Relation to other functions

[edit] Plots

[edit] Particular values

[edit] Approximations

[edit] Applications

[edit] See also

[edit] References and further reading

[edit] External links

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