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 $\mathrm{\Gamma }(n)=(n-1)!$

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 a positive integer, the function is defined by:

$\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{e}^{-t}\mathrm{d}t.$

Properties

Particular values

Some particular values of the gamma function are:

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

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

$\mathrm{\Pi }(z)=\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{t}^{z+1}\frac{\mathrm{d}t}{t},$

so that

$\mathrm{\Pi }(n)=n!$

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:

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

Bernhard Riemann found a relation between these two functions. This was 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")

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

Related pages

• Gamma distribution

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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