Gamma function

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

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 ${\displaystyle \Gamma (n)=(n-1)!}$[1][2]

The gamma function along part of the real axis

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:[2][3]

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

Properties

Particular values

Some particular values of the gamma function are:

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

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

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

so that

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

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:

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

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

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

Notes

1. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-10-05.
2. Weisstein, Eric W. "Gamma Function". mathworld.wolfram.com. Retrieved 2020-10-05.
3. "gamma function | Properties, Examples, & Equation". Encyclopedia Britannica. Retrieved 2020-10-05.

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)
• 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. CS1 maint: discouraged parameter (link)
• 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.