General
The gamma function 
is applied in exact 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 this argument. This relation is described by the formula:
Euler derived some basic properties and formulas for the gamma function. He started investigations of 
from the infinite product:
The gamma function 
has a long history of development and numerous applications since 1729 when Euler derived his famous integral representation of the factorial function. In modern notation it can be rewritten as the following:
The history of the gamma function is described in the subsection "General" of the section "Gamma function." Since the famous work of J. Stirling (1730) who first used series for 
to derive the asymptotic formula for 
, mathematicians have used the logarithm of the gamma function 
for their investigations of the gamma function 
. Investigators of mention include: C. Siegel, A. M. Legendre, K. F. Gauss, C. J. Malmstén, O. Schlömilch, J. P. M. Binet (1843), E. E. Kummer (1847), and G. Plana (1847). M. A. Stern (1847) proved convergence of the Stirling's series for the derivative of 
. C. Hermite (1900) proved convergence of the Stirling's series for 
if 
is a complex number.
During the twentieth century, the function log(Γ(z)) was used in many works where the gamma function was applied or investigated. The appearance of computer systems at the end of the twentieth century demanded more careful attention to the structure of branch cuts for basic mathematical functions to support the validity of the mathematical relations everywhere in the complex plane. This lead to the appearance of a special log‐gamma function 
, which is equivalent to the logarithm of the gamma function 
as a multivalued analytic function, except that it is conventionally defined with a different branch cut structure and principal sheet. The log‐gamma function 
was introduced by J. Keiper (1990) for Mathematica. It allows a concise formulation of many identities related to the Riemann zeta function 
.
The importance of the gamma function and its Euler integral stimulated some mathematicians to study the incomplete Euler integrals, which are actually equal to the indefinite integral of the expression 
. They were introduced in an article by A. M. Legendre (1811). Later, P. Schlömilch (1871) introduced the name "incomplete gamma function" for such an integral. These functions were investigated by J. Tannery (1882), F. E. Prym (1877), and M. Lerch (1905) (who gave a series representation for the incomplete gamma function). N. Nielsen (1906) and other mathematicians also had special interests in these functions, which were included in the main handbooks of special functions and current computer systems like Mathematica.
The needs of computer systems lead to the implementation of slightly more general incomplete gamma functions and their regularized and inverse versions. In addition to the classical gamma function 
, Mathematica includes the following related set of gamma functions: incomplete gamma function 
, generalized incomplete gamma function 
, regularized incomplete gamma function 
, generalized regularized incomplete gamma function 
, log‐gamma function 
, inverse of the regularized incomplete gamma function 
, and inverse of the generalized regularized incomplete gamma function 
.
Definitions of gamma functions
The gamma function 
, the incomplete gamma function 
, the generalized incomplete gamma function 
, the regularized incomplete gamma function 
, the generalized regularized incomplete gamma function 
, the log‐gamma function (almost equal to the logarithm of the gamma function) 
, the inverse of the regularized incomplete gamma function 
, and the inverse of the generalized regularized incomplete gamma function 
are defined by the following formulas:
The function
is equivalent to
as a multivalued analytic function, except that it is conventionally defined with a different branch cut structure and principal sheet. The function
allows a concise formulation of many identities related to the Riemann zeta function
:
The previous functions comprise the interconnected group called the gamma functions.
Instead of the first three previous classical definitions using definite integrals, the other equivalent definitions with infinite series can be used.
Connections within the group of gamma functions and with other function groups
The incomplete gamma functions 
, 
, 
, and 
are particular cases of the more general hypergeometric and Meijer G functions.
For example, they can be represented through hypergeometric functions 
and 
or the Tricomi confluent hypergeometric function 
:
These functions also have rather simple representations in terms of classical Meijer G functions:
The log‐gamma function 
can be expressed through polygamma and zeta functions by the following formulas:
The gamma functions 
, 
, 
, and 
can be represented using the related exponential integral 
by the following formulas:
The gamma functions 
, 
, 
, and 
are connected with the inverse of the regularized incomplete gamma function 
and the inverse of the generalized regularized incomplete gamma function 
by the following formulas:
The gamma functions 
, 
, 
, 
, 
, and 
are connected with each other by the formulas:
The best-known properties and formulas for exponential integrals
For real values of 
, the values of the gamma function 
are real (or infinity). For real values of the parameter 
and positive arguments 
, 
, 
, the values of the gamma functions 
, 
, 
, 
, and 
are real (or infinity).
The gamma functions 
, 
, 
, 
, 
, 
, 
, and 
have the following values at zero arguments:
If the variable 
is equal to 
and 
, the incomplete gamma function 
coincides with the gamma function 
and the corresponding regularized gamma function 
is equal to 
:
In cases when the parameter 
equals 
, the incomplete gamma functions 
and 
can be expressed as an exponential function multiplied by a polynomial. In cases when the parameter 
equals 
, the incomplete gamma function 
can be expressed with the exponential integral 
, exponential, and logarithmic functions, but the regularized incomplete gamma function 
is equal to 
. In cases when the parameter 
equals 
the incomplete gamma functions 
and 
can be expressed through the complementary error function 
and the exponential function, for example:
These formulas are particular cases of the following general formulas:
If the argument 
, the log‐gamma function 
can be evaluated at these points where the gamma function can be evaluated in closed form. The log‐gamma function 
can also be represented recursively in terms of 
for 
:
The generalized incomplete gamma functions 
and 
in particular cases can be represented through incomplete gamma functions 
and 
and the gamma function 
:
The inverse of the regularized incomplete gamma functions 
and 
for particular values of arguments satisfy the following relations:
The gamma functions 
, 
, 
, 
, 
, and 
are defined for all complex values of their arguments.
The functions 
and 
are analytic functions of 
and 
over the whole complex 
‐ and 
‐planes excluding the branch cut on the 
‐plane. For fixed 
, they are entire functions of 
. The functions 
and 
are analytic functions of 
, 
, and 
over the whole complex 
‐, 
‐, and 
‐planes excluding the branch cuts on the 
‐ and 
‐planes. For fixed 
and 
, they are entire functions of 
.
The function 
is an analytical function of 
over the whole complex 
‐plane excluding the branch cut.
For fixed 
, the functions 
and 
have an essential singularity at 
. At the same time, the point 
is a branch point for generic 
. For fixed 
, the functions 
and 
have only one singular point at 
. It is an essential singularity.
For fixed 
, the functions 
and 
have an essential singularity at 
(for fixed 
) and at 
(for fixed 
). At the same time, the points 
are branch points for generic 
. For fixed 
and 
, the functions 
and 
have only one singular point at 
. It is an essential singularity.
The function 
does not have poles or essential singularities.
For fixed 
, not a positive integer, the functions 
and 
have two branch points: 
and 
.
For fixed 
, not a positive integer, the functions 
and 
are single‐valued functions on the 
‐plane cut along the interval 
, where they are continuous from above:
For fixed 
, the functions 
and 
do not have branch points and branch cuts.
For fixed 
, 
or fixed 
, 
(with 
), the functions 
and 
have two branch points with respect to 
or 
: 
, 
.
For fixed 
and 
, the functions 
and 
are single‐valued functions on the 
‐plane cut along the interval 
, where they are continuous from above:
For fixed 
and 
, the functions 
and 
are single‐valued functions on the 
‐plane cut along the interval 
, where they are continuous from above:
For fixed 
and 
, the functions 
and 
do not have branch points and branch cuts.
The function 
has two branch points: 
and 
.
The function 
is a single‐valued function on the 
‐plane cut along the interval 
, where it is continuous from above:
The gamma functions 
, 
, 
, 
, 
, the log‐gamma function 
, and their inverses 
and 
do not have periodicity.
The gamma functions 
, 
, 
, 
, and the log‐gamma function 
have mirror symmetry (except on the branch cut intervals):
Two of the gamma functions have the following permutation symmetry:
The gamma functions 
, 
, 
, 
, the log‐gamma function 
, and the inverse 
have the following series expansions:
The asymptotic behavior of the gamma functions 
and 
, the log‐gamma function 
, and the inverse 
can be described by the following formulas (only the main terms of asymptotic expansion are given):
The gamma functions 
, 
, 
, 
, and the log‐gamma function 
can also be represented through the following integrals:
The argument of the log‐gamma function 
can be simplified if 
or 
:
The log‐gamma function 
with 
can be represented by a formula that follows from the corresponding multiplication formula for the gamma function 
:
The gamma functions 
, 
, 
, 
, and the log‐gamma function 
satisfy the following recurrence identities:
The previous formulas can be generalized to the following recurrence identities with a jump of length n:
The derivatives of the gamma functions 
, 
, 
, and 
with respect to the variables 
, 
, and 
have simple representations in terms of elementary functions:
The derivatives of the log‐gamma function 
and the inverses of the regularized incomplete gamma functions 
, and 
with respect to the variables 
, 
, and 
have more complicated representations by the formulas:
The derivative of the exponential integral 
by its parameter 
can be represented in terms of the regularized hypergeometric function 
:
The derivatives of the gamma functions 
, 
, 
, and 
, and their inverses 
and 
with respect to the parameter 
can be represented in terms of the regularized hypergeometric function 
:
The symbolic 
-order derivatives of all gamma functions 
, 
, 
, 
, and their inverses 
, and 
have the following representations:
The gamma functions 
, 
, 
, and 
satisfy the following second-order linear differential equations:
where 
and 
are arbitrary constants.
The log‐gamma function 
satisfies the following simple first-order linear differential equation:
The inverses of the regularized incomplete gamma functions 
and 
satisfy the following ordinary nonlinear second-order differential equation:
Applications of gamma functions
The gamma functions are used throughout mathematics, the exact sciences, and engineering. In particular, the incomplete gamma function is used in solid state physics and statistics, and the logarithm of the gamma function is used in discrete mathematics, number theory, and other fields of sciences.
along the real axis. The real parts are shown in red and the imaginary parts are shown in blue.
along the real axis.
over the 
‐
–plane.
over the 
‐
-plane.
