Gamma function

The gamma function along part of the real axis

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

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:

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.

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")

Related pages

• Gamma distribution

References

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• Andrews, George E.. Special Functions (2000)Cambridge University Press. ISBN 978-0-521-78988-2.
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• P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
• Bonnar, James. The Gamma Function (2010)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. Numerical Recipes: The Art of Scientific Computing (2007)Cambridge University Press. ISBN 978-0-521-88068-8. Retrieved 2013-02-10.
• O. R. Rocktaeschel, ´´Methoden zur Berechnung der Gammafunktion für komplexes Argument``, University of Dresden, Dresden, 1922.
• Temme, Nico M.. Special Functions: An Introduction to the Classical Functions of Mathematical Physics (1996)John Wiley & Sons. ISBN 978-0-471-11313-3.
• Whittaker, E.T.. A Course of Modern Analysis (1996)Cambridge University Press. ISBN 978-0-521-58807-2.