gamma function

The function itself was initially defined as an integral (in modern representation, ${\displaystyle \textstyle \Gamma (x)=\int _{0}^{\infty }e^{-t}t^{x-1}dt}$) for positive real x by Swiss mathematician Leonhard Euler in 1730. The name derives from the notation, Γ(x), which was introduced by Adrien-Marie Legendre (1752—1833) (he referred to it, however, as the Eulerian integral of the second kind). Both Euler's integral and Legendre's notation shift the argument with respect to the factorial, so that for integer n>0, Γ(n) = (n−1)!. Carl Friedrich Gauss preferred π(x), with no shift, but Legendre's notation prevailed. Generalisation to non-integer negative and to complex numbers was achieved by analytic continuation.^[1]

gamma function (plural gamma functions)

1. (mathematical analysis) A meromorphic function which generalizes the notion of factorial to complex numbers and has singularities at the nonpositive integers; any of certain generalizations or analogues of said function, such as extend the factorial to domains other than the complex numbers.
   • 1987, Kit Ming Yeung, Applications of p-adic gamma function to congruences of binomial coefficients, University of California, San Diego, page 3:

     Chapter 3 deals with the p-adic gamma function.

   • 2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman & Hall / CRC Press, page 2,

     In particular, the exponential, circular, and hyperbolic functions are rational combinations of gamma functions.

   • 2007, Philip J. Davis, “Leonhard Euler's Integral: A Historical Profile of the Gamma Function”, in William Dunham, editor, The Genius of Euler: Reflections on His Life and Work, American Mathematical Society, page 167:

     We select one mathematical object, the gamma function, and show how it grew in concept and in content from the time of Euler to the recent mathematical treatise of Bourbaki, and how, in this growth, it partook of the general development of mathematics over the past two and a quarter centuries.

• (function that extends the domain of the factorial): Euler integral of the second kind (regarded as an integral)

• function

• digamma function
• incomplete gamma function
• polygamma function
• trigamma function

• gamma function on Wikipedia.Wikipedia
• Euler integral on Wikipedia.Wikipedia
• p-adic gamma function on Wikipedia.Wikipedia