In mathematics, the gamma function (Γ(z)) is a key topic in the field of special functions. Γ(z) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as [1][2]
The gamma function is defined for all complex numbers, but it is not defined for negative integers and zero. For a complex number whose real part is a positive integer, the function is defined by:[2][3]
Properties
Particular values
Some particular values of the gamma function are:
Pi function
Gauss introduced the Pi function. This is another way of denoting the gamma function. In terms of the gamma function, the Pi function is
so that
for every non-negative integer n.
Applications
Analytic number theory
The gamma function is used to study the Riemann zeta function. A property of the Riemann zeta function is its functional equation:
Bernhard Riemann found a relation between these two functions. This was published in his 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Prime Numbers less than a Given Quantity")
Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6)
Emil Artin, "The Gamma Function", in Rosen, Michael (ed.) Exposition by Emil Artin: a selection; History of Mathematics 30. Providence, RI: American Mathematical Society (2006).
P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
Bonnar, James (2010). The Gamma Function. Createspace Independent Publishing. ISBN978-1-4636-9429-6.
Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," American Mathematical Monthly66, 849-869 (1959)
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.1. Gamma Function", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8, archived from the original on 2021-10-27, retrieved 2013-02-10
O. R. Rocktaeschel, ´´Methoden zur Berechnung der Gammafunktion für komplexes Argument``, University of Dresden, Dresden, 1922.
Temme, Nico M. (1996). Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons. ISBN978-0-471-11313-3.
Whittaker, E.T.; Watson, G.N. (1996). A Course of Modern Analysis. Cambridge University Press. ISBN978-0-521-58807-2.