Gamma function

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The gamma function along part of the real axis

In mathematics, the gamma function (Γ(z)) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as

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 not a negative integer, the function is defined by:

Properties[change | change source]

Particular values[change | change source]

Some particular values of the gamma function are:

Pi function[change | change source]

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[change | change source]

Analytic number theory[change | change source]

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 in 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Prime Numbers less than a Given Quantity")

Notes[change | change source]

References[change | change source]

  • Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6)
  • G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 2001. ISBN 978-0-521-78988-2. Chapter one, covering the gamma and beta functions, is highly readable and definitive.
  • 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).
  • Birkhoff, George D. (1913). "Note on the gamma function". Bull. Amer. Math. Soc. 20 (1): 1–10.
  • P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
  • James D. Bonnar, The Gamma Function. CreateSpace Publishing, Seattle, 2010. ISBN 978-1463694296. A thorough and systematic book devoted entirely to the subject of the gamma function.
  • Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," American Mathematical Monthly 66, 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, ISBN 978-0-521-88068-8 
  • O. R. Rocktaeschel, ´´Methoden zur Berechnung der Gammafunktion für komplexes Argument``, University of Dresden, Dresden, 1922.
  • Nico M. Temme, "Special Functions: An Introduction to the Classical Functions of Mathematical Physics", John Wiley & Sons, New York, ISBN 0-471-11313-1,1996.
  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge University Press (1927; reprinted 1996) ISBN 978-0521588072