Gamma function

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 $\Gamma (n)=(n-1)!$

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:

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,{\rm {d}}t.$

Properties

Particular values

Some particular values of the gamma function are:

${\begin{array}{lll}\Gamma (-3/2)&={\tfrac {4}{3}}{\sqrt {\pi }}&\approx 2.363271801207\\\Gamma (-1/2)&=-2{\sqrt {\pi }}&\approx -3.544907701811\\\Gamma (1/2)&={\sqrt {\pi }}&\approx 1.772453850905\\\Gamma (1)&=0!&=1\\\Gamma (3/2)&={\tfrac {1}{2}}{\sqrt {\pi }}&\approx 0.88622692545\\\Gamma (2)&=1!&=1\\\Gamma (5/2)&={\tfrac {3}{4}}{\sqrt {\pi }}&\approx 1.32934038818\\\Gamma (3)&=2!&=2\\\Gamma (7/2)&={\tfrac {15}{8}}{\sqrt {\pi }}&\approx 3.32335097045\\\Gamma (4)&=3!&=6\\\end{array}}$

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

$\Pi (z)=\Gamma (z+1)=z\;\Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z+1}\,{\frac {{\rm {d}}t}{t}},$

so that

$\Pi (n)=n!\,,$

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:

$\Gamma \left({\frac {s}{2}}\right)\zeta (s)\pi ^{-s/2}=\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)\pi ^{-(1-s)/2}.$

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")

$\zeta (z)\;\Gamma (z)=\int _{0}^{\infty }{\frac {t^{z}}{e^{t}-1}}\;{\frac {dt}{t}}.$

References

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6)
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• 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).
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• P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
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• Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," American Mathematical Monthly 66, 849-869 (1959)
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• O. R. Rocktaeschel, ´´Methoden zur Berechnung der Gammafunktion für komplexes Argument``, University of Dresden, Dresden, 1922.
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