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Gamma, Beta, Erf > Gamma[a,z] > Introduction to the gamma functions


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 .