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Chapter 2:
The Gamma Function and Related Functions
2.1 Introduction In the eighteenth century, L. Euler (1707–1783) concerned himself with the problem of interpolating between the numbers n!=∫ ∞ 0 e −t t n dtn=0,1,2,… with nonintegral values of n. This problem led Euler in 1729 to the now famous gamma function, a generalization of the factorial function that gives meaning to x! when x is any positive number. His result can be extended to certain negative numbers and even to complex numbers. The notation Γ(x) that is now widely accepted for the gamma function is not due to Euler, however, but was introduced in 1809 by A. Legendre (1752–1833), who was also responsible for the duplication formula for the gamma function. Nearly 150 years after Euler's discovery of it, the theory concerning the gamma function was greatly expanded by means of the theory of entire functions developed by K. Weierstrass (1815–1897). Because it is a generalization of n!, the gamma function has been examined over the years as a means of generalizing certain functions, operations, etc., that are commonly defined in terms of factorials. In addition, the gamma function is useful in the evaluation of many nonelementary integrals and in the definition of other special functions. Another function useful in various applications is the related beta function, often called the eulerian integral of the first kind.
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