Lerch (1905) (who gave a series representation for the incomplete gamma function). Schlömilch (1871) introduced the name "incomplete gamma function" for such an integral. 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. It allows a concise formulation of many identities related to the Riemann zeta function. The log‐gamma function was introduced by J. 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 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. Hermite (1900) proved convergence of the Stirling's series for if is a complex number.ĭuring the twentieth century, the function log(Γ(z)) was used in many works where the gamma function was applied or investigated. Stern (1847) proved convergence of the Stirling's series for the derivative of. 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. The history of the gamma function is described in the subsection "General" of the section "Gamma function." Since the famous work of J. In modern notation it can be rewritten as the following: 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. He started investigations of from the infinite product: This relation is described by the formula:Įuler derived some basic properties and formulas for the gamma function. Euler (1729) as a natural extension of the factorial operation from positive integers to real and even complex values of this argument. It was introduced by the famous mathematician L. The gamma function is applied in exact sciences almost as often as the well‐known factorial symbol. Gamma, Beta, Erf Gamma Introduction to the gamma functions
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