Introduction to the Gamma Function

General

The gamma function is used in the mathematical and applied sciences almost as often as the well-known factorial symbol . It was introduced by the famous mathematician L. Euler (1729) as a natural extension of the factorial operation from positive integers to real and even complex values of the argument . This relation is described by the following formula:

L. Euler derived some basic properties and formulas for the gamma function. He started investigations of from the infinite product

and the integral

which is currently known as the beta function integral. As a result, Euler derived the following integral representation for factorial :

which can be easily converted into the well-known Euler integral for the gamma function:

Also, during his research, Euler closely approached the famous reflection formula:

which later got his name.

At the same time, J. Stirling (1730) found the famous asymptotic formula for the factorial, which bears his name. This formula was also naturally applied to the gamma function resulting in the following asymptotic relation:

Later, A. M. Legendre (1808, 1814) suggested the current symbol Γ for the gamma function and discovered the duplication formula:

It was generalized by C. F. Gauss (1812) to the multiplication formula:

F. W. Newman (1848) studied the reciprocal of the gamma function and found that it is an entire function and has the following product representation valid for the whole complex plane:

where is the Euler–Mascheroni gamma constant.

B. Riemann (1856) proved an important relation between the gamma and zeta functions:

which was mentioned centuries ago in an article by Euler (1749) for particular values of the argument .

K. Weierstrass (1856) and other nineteenth century mathematicians widely used the gamma function in their investigations and discovered many more complicated properties and formulas for it. In particular, H. Hankel (1864, 1880) derived its contour integral representation for complex arguments, and O. Hölder (1887) proved that the gamma function does not satisfy any algebraic differential equation. This result was subsequently re-proved by A. Ostrowski (1925).

Many mathematicians devote special attention to the question of the uniqueness of extending the factorial operation from positive integers to arbitrary real or complex values. Evidently this question is connected to the solutions of the functional equation:

J. Hadamard (1894) found that the function is an entire analytic function that coincides with for . But this function satisfies the more complicated functional equation and has a more complicated integral representation than the classical gamma function defined by the Euler integral.

H. Bohr and J. Mollerup (1922) proved that the gamma function is the only function that satisfies the recurrence relationship , is positive for , equals one at , and is logarithmically convex (that is, is convex). If the restriction on convexity is absent, then the recurrence relationship has an infinite set of solutions in the form , where is an arbitrary periodic function with period .

Definition of gamma function

The gamma function in the half-plane is defined as the value of the following definite integral:

This integral is an analytic function that can be represented in different forms; for example, as the following sum of an integral and a series without any restrictions on the argument:

The last formula can also be used as an equivalent definition of the gamma function.

A quick look at the gamma function

Here is a quick look at the graphics for the gamma function along the real axis.

Connections within the group of gamma functions and with other function groups

Representations through more general functions

The gamma function is the main example of a group of functions collectively referred to as gamma functions. For example, it can be written in terms of the incomplete gamma function:

All four incomplete gamma functions , , , and can be represented as cases of the hypergeometric function . Further, the gamma function Γ(z) is the special degenerate case of the hypergeometric function .

Representations through related equivalent functions

The gamma function and two factorial functions are connected by the formulas:

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:

Analyticity

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.

Periodicity

The function does not have periodicity.

Parity and symmetry

The function has mirror symmetry:

Differentiation

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

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:

Integral representations

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 .

Product representations

The following infinite product representation for clearly illustrates that at :

The similar product representation for illustrates that is an entire function:

Limit representations

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:

Definite integration

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:

Integral transforms

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:

Transformations

The following formulas describe some transformations of the gamma functions with linear arguments into expressions that contain the gamma function with the simplest argument:

Multiple arguments

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:

Identities

The gamma function satisfies the following recurrence identities:

These formulas can be generalized to the following recurrence identities with a jump of length :

Inequalities

The most famous inequalities for the gamma function can be described by the following formulas:

Applications

The gamma function is used throughout mathematics, the exact sciences, and engineering.