Gamma function

extension of the factorial function, with its argument shifted down by 1, to real and complex numbers
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:

PropertiesEdit

Particular valuesEdit

Some particular values of the gamma function are:

 

Pi functionEdit

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.

ApplicationsEdit

Analytic number theoryEdit

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")

 

NotesEdit

ReferencesEdit

  • Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6)
  • Andrews, George E.; Askey, Richard; Roy, Ranjan (2000). Special Functions. Cambridge University Press. ISBN 978-0-521-78988-2.
  • 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. MR 1559418.
  • P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
  • Bonnar, James (2010). The Gamma Function. Createspace Independent Publishing. ISBN 978-1-4636-9429-6.
  • 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.
  • Temme, Nico M. (1996). Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons. ISBN 978-0-471-11313-3.
  • Whittaker, E.T.; Watson, G.N. (1996). A Course of Modern Analysis. Cambridge University Press. ISBN 978-0-521-58807-2.