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The gamma function  can be exactly evaluated in the points  . Here are examples:
The preceding evaluations can be provided by the formulas:
At the points  , the values of the gamma function  can be represented through values of  :
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.
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.
The function  does not have branch points and branch cuts.
The function  does not have periodicity.
The function  has mirror symmetry:
The derivatives of  can be represented through gamma and polygamma functions:
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 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:
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.
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