The best-known properties and formulas for the gamma function
Values at points
The gamma function can be exactly evaluated in the points . Here are examples:
Specific values for specialized variables
The preceding evaluations can be provided by the formulas:
At the points , the values of the gamma function can be represented through values of :
Real values for real arguments
For real values of argument , the values of the gamma function are real (or infinity). The gamma function is not equal to zero:
The gamma function is an analytical function of , which is defined over the whole complex ‐plane with the exception of countably many points . The reciprocal of the gamma function is an entire function.
Poles and essential singularities
The function has an infinite set of singular points , which are the simple poles with residues . The point is the accumulation point of the poles, which means that is an essential singular point.
Branch points and branch cuts
The function does not have branch points and branch cuts.
The function does not have periodicity.
Parity and symmetry
The function has mirror symmetry:
The derivatives of can be represented through gamma and polygamma functions:
Ordinary differential equation
The gamma function does not satisfy any algebraic differential equation (O. Hölder, 1887). But it is the solution of the following nonalgebraic equation:
Series representations of the gamma function near the poles are of great interest for applications in the theory of generalized hypergeometric, Meijer G, and Fox H functions. These representations can be described by the formulas:
where are the Bernoulli numbers.
Asymptotic series expansions
Asymptotic behavior of the gamma function is described by the famous Stirling formula:
This formula allows derivation of the following asymptotic expansion for the ratio of gamma functions:
The gamma function has several integral representations that are different from the Euler integral:
and related integral
which can be used for defining the gamma function over the whole complex plane.
Some of the integral representations are the following:
This final formula is known as Hankel's contour integral. The path of integration starts at on the real axis, goes to , circles the origin in the counterclockwise direction with radius to the point , and returns to the point .
The following infinite product representation for clearly illustrates that at :
The similar product representation for illustrates that is an entire function:
The following famous limit representation for was known to L. Euler:
It can be modified to the following related limit representations:
The gamma function can be evaluated as the limit of the following definite integral:
The most famous definite integrals, including the gamma function, belong to the class of Mellin–Barnes integrals. They are used to provide a uniform representation of all generalized hypergeometric, Meijer G, and Fox H functions. For example, the Meijer G function is defined as the value of the following Mellin–Barnes integral:
The infinite contour of integration separates the poles of at , from the poles of at , . Such a contour always exists in the cases .
There are three possibilities for the contour :
(i) runs from γ-ⅈ∞ to γ+ⅈ∞ (where ) so that all poles of , , are to the left of , and all the poles of , are to the right of ℒ. This contour can be a straight line if (then ). In this case, the integral converges if , . If , then must be real and positive, and the additional condition , should be added.
(ii) is a left loop, starting and ending at and encircling all poles of ,, once in the positive direction, but none of the poles of , . In this case, the integral converges if and either or and or and and and .
(iii) is a right loop, starting and ending at +∞ and encircling all poles of , , once in the negative direction, but none of the poles of , . In this case, the integral converges if , and either or and or and and and .
In particular cases, the last integral can be evaluated using simpler elementary and special functions:
The definition of the Meijer G function through a Mellin‐Barnes integral realizes the inverse Mellin integral transform of ratios of gamma functions:
The contour is the vertical straight line . It allows the writing of the following rather general formula for the inverse Mellin integral transform:
In particular cases, it gives the following representations:
The following formulas describe some transformations of the gamma functions with linear arguments into expressions that contain the gamma function with the simplest argument:
In the case of multiple arguments , ,…, , the gamma function can be represented by the following duplication and multiplication formulas, derived by A. M. Legendre and C. F. Gauss:
Products involving the direct function
The product of two gamma functions and , with arguments satisfying the condition that is an integer, can be represented through elementary functions:
The preceding formula transforms into the following formula and its relatives:
The ratio of two gamma functions and , with arguments satisfying the condition that is integer, can be represented through a polynomial or rational function:
The gamma function satisfies the following recurrence identities:
These formulas can be generalized to the following recurrence identities with a jump of length :
The most famous inequalities for the gamma function can be described by the following formulas:
The gamma function is used throughout mathematics, the exact sciences, and engineering.