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

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

but for non-natural 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 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

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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^{z-1}\,e^{-t}\,dt converges absolutely. Using integration by parts, one can show that

$\backslash Gamma(z+1)=z\; \backslash ,\; \backslash Gamma(z)\backslash ,.$

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

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

for all natural numbers 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

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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)}

$\backslash Gamma(z)\; =\; \backslash frac\{e^\{-\backslash gamma\; z\}\}\{z\}\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(1\; +\; \backslash frac\{z\}\{n\}\backslash right)^\{-1\}\; e^\{z/n\}$

where $\backslash gamma$ is the Euler-Mascheroni constant.

Properties

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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

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

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 $\backslash sqrt\; \backslash pi$.

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

$\backslash Gamma\text{'}(z)=\backslash Gamma(z)\backslash psi\_0(z).\backslash ,$

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

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

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

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

so that

$\backslash Pi(n)\; =\; n!.\backslash ,$

Using the Pi function the reflection formula takes on the form

$\backslash Pi(z)\; \backslash ;\; \backslash Pi(-z)\; =\; \backslash frac\{\backslash pi\; z\}\{\backslash sin(\; \backslash pi\; z)\}\; =\; \backslash frac\{1\}\{\backslash mathrm\{sinc\}(z)\}$

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

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

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.

Relation to other functions

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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.

Plots

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Image:Gamma real.png|Real part of Γ(z) Image:Gamma imag.png|Imaginary part of Γ(z) Image:Gamma absolute.png|Absolute value of Γ(z)

Image:Log gamma real.png|Real part of log Γ(z) Image:Log gamma imag.png|Imaginary part of log Γ(z) Image:Log gamma absolute.png|Absolute value of log Γ(z)

Particular values

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Main article: Particular values of the Gamma function

{| $\backslash Gamma(-3/2)\backslash ,$ $=\; \backslash frac\; \{4\backslash sqrt\{\backslash pi\}\}\; \{3\}\; \backslash ,$ $\backslash Gamma(-1/2)\backslash ,$ $=\; -2\backslash sqrt\{\backslash pi\}\backslash ,$ $\backslash Gamma(1/2)\backslash ,$ $=\; \backslash sqrt\{\backslash pi\}\backslash ,$ $\backslash Gamma(1)\backslash ,$ $=0!=1\; \backslash ,$ $\backslash Gamma(3/2)\backslash ,$ $=\; \backslash frac\; \{\backslash sqrt\{\backslash pi\}\}\; \{2\}\; \backslash ,$ $\backslash Gamma(2)\backslash ,$ $=1!=1\; \backslash ,$ $\backslash Gamma(5/2)\backslash ,$ $=\; \backslash frac\; \{3\; \backslash sqrt\{\backslash pi\}\}\; \{4\}\; \backslash ,$ $\backslash Gamma(3)\backslash ,$ $=2!=2\; \backslash ,$ $\backslash Gamma(7/2)\backslash ,$ $=\; \backslash frac\; \{15\backslash sqrt\{\backslash pi\}\}\; \{8\}\; \backslash ,$ $\backslash Gamma(4)\backslash ,$ $=3!=6\; \backslash ,$

Approximations

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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

$\backslash Gamma(z)\; =\; x^z\; e^\{-x\}\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^n\}\{z(z+1)\; \backslash dots\; (z+n)\}\; +\; \backslash int\_x^\backslash infty\; e^\{-t\}\; t^\{z-1\}\; dt$

where, if Re(z) has been reduced to the interval 2, the last integral is smaller than . 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 $O((\backslash log\; N)^2\; M(N))$ 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).

See also

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• 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

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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)

• 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

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• Examples of problems involving the Gamma function can be found at Exampleproblems.com.
• Gamma function calculator
• Wolfram gamma function evaluator (arbitrary precision)

Gamma and related functions | Special hypergeometric functions

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