Gamma Function


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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 ?
An important fact for the gamma function in applied mathematics as well as in probability is the Stirling formula that gives a pretty accurate idea about the size of the gamma function.
Davis: Gamma Function and Related Functions, in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, (M.
Assumption x < y < z for any 3 points of the gamma function, if [GAMMA](x) < [GAMMA](y) < [GAMMA](z) (or [GAMMA](x) > [GAMMA](y) > [GAMMA](z)) established, then the point of y is in increasing interval of the gamma function.
On the q-analogue of gamma functions and related inequalities.
The argument uses an asymptotic formula for the Gamma function, a special Mellin transform and one of the Phragmen-Lindelof principles.
In the first case, the gamma function was used as before but with some adjustment to the monomer molar mass and the shape factor.
Integration Techniques, Gamma Function, Beta Function, Cauchy
x] is the drop concentration per unit volume (m-3), and G the gamma function with [[alpha].
Rudert, Tables of the Incomplete Gamma Function Ratio, Justus von Liebig Verlag, Darmstadt, Germany (1965).
In Equation (2), [Gamma] (a) is the gamma function, the integral is the incomplete gamma function denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
For the professional and prosumer customers, the [eth]7II adds picture profiles for convenient in-camera tone adjustments and supports the S-Log2 gamma function which preserves a wide dynamic range.
Now using Legendre's duplication formula and recurrence relation for Gamma function, the above formula can be written in the form