# Gamma Function

Also found in: Dictionary, Wikipedia.
Related to Gamma Function: Incomplete gamma function

## gamma function

[′gam·ə ‚fəŋk·shən]
(mathematics)
The complex function given by the integral with respect to t from 0 to ∞ of e -t t z-1; this function helps determine the general solution of Gauss' hypergeometric equation.

## Gamma Function

Γ(x), one of the most important special functions; generalizes the concept of the factorial. For all positive n it is given by Γ(n) = (n - 1)! = 1·2 … (n - 1). It was first introduced by L. Euler in 1729. For real values of x > 0 it is defined by the equality Another notation is

Γ(x + 1) = π(x) = x!

The principal relations for the gamma function are

Γ(x + 1) = xΓ(x) (functional equation)

Γ(x)Γ(1 - x) = π/sin πx (complementary formula) Special values are For large x the Stirling formula holds: A large number of definite integrals, infinite products, and summations of series are expressed by the gamma function. The function has also been extended to complex values of the independent variable.

### REFERENCES

Janke, E., and F. Emde. Tablitsy funktsii s formulami i krivymi, 3rd ed. Moscow, 1959. (Translated from German.)
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 6th ed., vol. 2. Moscow, 1966.
References in periodicals archive ?
1] ([alpha], u) is the inverse function of regularized incomplete gamma function.
The gamma function has many beautiful properties and has been used in almost all the branches of science and engineering.
which should not be confused with the standard gamma function [LAMBDA](x) = [[integral].
Now, using the duplication formula for gamma function, namely,
The MGF approach along with the Gauss's hyper-geometric, Gauss's hyperbolic and Gamma functions has been used, to eliminate these kinds of integration difficulties.
Gamma Function is complex but it is necessary to know their values only over a fairly narrow range of values, corresponding to the standard deviation of the wind, respectively at 20 and 100% of average speed.
where b is neither zero nor negative integer and the notation [GAMMA] stands for gamma function.
Many of the classical facts for the gamma function have been extended to the q-gamma function which is defined as
The NB log-likelihood, parameterized in terms of the vector of model coefficients [beta], can be expressed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [alpha] is the dispersion parameter, x is a vector of covariates, [GAMMA](*) is the gamma function, and i= 1, 2, .
Let c denote a non-zero constant, p(z) a non-zero polynomial, and [GAMMA](z) the Gamma function.
This last identity is a consequence of the duplication formula for the Gamma function, [GAMMA](r)[GAMMA](r + 1/2) = [2.

Site: Follow: Share:
Open / Close