NationMaster - Encyclopedia: Gamma function

In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by Image File history File links Gamma_plot. ... Image File history File links Gamma_plot. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Gamma (uppercase Î“, lowercase Î³) is the third letter of the Greek alphabet. ... For factorial rings in mathematics, see unique factorisation domain. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

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

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. The Gamma function generalizes the factorial function for non-integer and complex values of n.

The Gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics. Probability is the likelihood that something is the case or will happen. ... This article is about the field of statistics. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

1 Definition
- 1.1 Main definition
- 1.2 Alternative definitions
2 Properties
3 Plots
4 Particular values
5 Approximations
6 See also
7 References
8 External links
- 8.1 Web sites
- 8.2 Further reading

Definition

Main definition

The extended version of the Gamma function in the complex plane

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 Image File history File links Metadata Size of this preview: 600 Ã— 600 pixelsFull resolution (731 Ã— 731 pixel, file size: 180 KB, MIME type: image/jpeg) This is the color function used in the picture above File historyClick on a date/time to view the file as it appeared at that... Image File history File links Metadata Size of this preview: 600 Ã— 600 pixelsFull resolution (731 Ã— 731 pixel, file size: 180 KB, MIME type: image/jpeg) This is the color function used in the picture above File historyClick on a date/time to view the file as it appeared at that... Adrien-Marie Legendre (September 18, 1752 â€“ January 10, 1833) was a French mathematician. ... This article is about the concept of integrals in calculus. ...

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

converges absolutely. Using integration by parts, one can show that In mathematics, a series is a sum of a sequence of terms. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

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

This functional equation generalizes relation n! = n×(n-1)! of the factorial function. We can evaluate Γ(1) analytically: In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

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

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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

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. Image File history File links Size of this preview: 800 Ã— 451 pixelsFull resolution (966 Ã— 544 pixel, file size: 200 KB, MIME type: image/png) Created using Mathematica: Plot3D[Abs[Gamma[x + I y]], {x, -4. ... Image File history File links Size of this preview: 800 Ã— 451 pixelsFull resolution (966 Ã— 544 pixel, file size: 200 KB, MIME type: image/png) Created using Mathematica: Plot3D[Abs[Gamma[x + I y]], {x, -4. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic 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: In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...

$begin{align} Gamma(z) &= lim_{n to infty} frac{n! ; n^z}{z ; (z+1)cdots(z+n)} Gamma(z) &= frac{e^{-gamma z}}{z} prod_{n=1}^infty left(1 + frac{z}{n}right)^{-1} e^{z/n} end{align}$

where γ is the Euler-Mascheroni constant. The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ...

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

$begin{align} Gamma(z+1) &= lim_{n to infty} frac{n! ; n^{z+1}}{(z+1) ; (z+2)cdots(z+1+n)} &= lim_{n to infty} left( z ; frac{n! ; n^z}{z ; (z+1) ; (z+2)cdots(z+n)} ; frac{n}{(z+1+n)}right) &= z ; Gamma(z) ; lim_{n to infty} frac{n}{(z+1+n)} &= z ; Gamma(z) end{align}$

Properties

General

Other important functional equations for the Gamma function are Euler's reflection formula In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a-x) and f(x). ...

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

and the duplication formula

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

The duplication formula is a special case of the multiplication theorem In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. ...

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

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

$Gammaleft(frac{1}{2}right)=sqrt{pi}, ,!$

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: In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...

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

where n!! denotes the double factorial. In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ...

The derivatives of the Gamma function are described in terms of the polygamma function. For example: In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ...

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

The Gamma function has a pole of order 1 at z = −n for every natural number n; the residue there is given by In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ...

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

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. In mathematical analysis, the Bohr_Mollerup theorem, named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it, characterizes the gamma function, defined for x > 0 by as the only function f on the interval x > 0 that simultaneously has the three properties and and is a convex function. ... A function is log-concave, if its natural log , is concave. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ...

$Gamma(z+1)=zGamma(z),$ because:

$begin{align} Gamma(z+1) &= int_0^infty t^{z+1-1}e^{-t},mathrm{d}t &= int_0^infty t^{z}e^{-t},mathrm{d}t end{align}$

And with integration by parts:

$begin{align} &= [ t^{z}frac{1}{ln(e^{-1})}(e^{-1})^{t} ]_{0}^{infty} + int_0^infty zt^{z-1}e^{-t},mathrm{d}t &= underbrace{[ -t^{z}e^{-t} ]_{0}^{infty}}_{=0-0} + int_0^infty zt^{z-1}e^{-t},mathrm{d}t &= zint_0^infty t^{z-1}e^{-t},mathrm{d}t &= zGamma(z) end{align}$

Pi function

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 Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

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

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 The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...

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

We also sometimes find

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

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. In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

Relation to other functions

In the first integral above, which defines the Gamma function, the limits of integration are fixed.

The incomplete Gamma function, Γ(a, x), is the function obtained by allowing either the upper or lower limit of integration, x, to vary. In mathematics, the gamma function is defined by a definite integral. ...

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).

$pi^{-z/2} ; Gammaleft(frac{z}{2}right) zeta(z) = pi^{-frac{1-z}{2}} ; Gammaleft(frac{1-z}{2}right) ; zeta(1-z).$

And also in the following elegant formula:

$zeta(z) ; Gamma(z) = int_{0}^{infty} frac{u^{z-1}}{e^u - 1} ; mathrm{d}u ,!.$ Which is only valid for Re(z) > 1.

In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ... In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ... In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ... In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. ... In mathematics, an exponential sum may be a finite Fourier series (i. ... Plot of 1/Î“(x) along the real axis In mathematics, the reciprocal Gamma function is the function where denotes the Gamma function. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

Plots

Real part of Γ(z) Download high resolution version (996x992, 279 KB)Real part of the Gamma function (colors cycling). ...

Imaginary part of Γ(z) Download high resolution version (996x992, 280 KB)Imaginary part of the Gamma function (colors cycling). ...

Absolute value of Γ(z) Download high resolution version (996x992, 227 KB)Absolute value of the Gamma function (colors cycling). ...

Real part of log Γ(z) Download high resolution version (996x992, 173 KB)Real part of the logarithm of the Gamma function (colors cycling). ...

Imaginary part of log Γ(z) Download high resolution version (996x992, 196 KB)Imaginary part of the logarithm of the Gamma function (colors cycling). ...

Absolute value of log Γ(z) Download high resolution version (996x992, 206 KB)Absolute value of the logarithm of the Gamma function (colors cycling). ...

Particular values

Main article: Particular values of the Gamma function The Gamma function is an important special function in mathematics. ...

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

Approximations

Complex values of the Gamma function can be computed numerically with arbitrary precision using Stirling's approximation or the Lanczos approximation. The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ... In mathematics, the Lanczos approximation is an approximation of the Gamma function published in 1964 by Cornelius Lanczos. ...

Applying integration by parts to Euler's integral, the Gamma function can also be written In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

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

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. In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. ...

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). In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ... The Gamma function is an important special function in mathematics. ...

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. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

References

^ 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)

Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," Am. Math. Monthly 66, 849-869 (1959)
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
Julian Havil, Gamma, Exploring Euler's Constant", ISBN 0-691-09983-9 (c) 2003

James Roy Newman was a mathematician and mathematical historian. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

Wikimedia Commons has media related to:

Gamma and related functions

Image File history File links Commons-logo. ...

Web sites

R. A. Askey, R. Roy, DLMF article about the Gamma function
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.
Computing the Gamma function - various algorithms

Richard Askey is an American mathematician. ... The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology to develop a major resource of math reference data for special functions and their applications. ... Wolfram Research is part of the Wolfram Group which consists of four companies: Wolfram Research Inc. ...

Contents

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