Gamma function

File:Gamma plot.svg

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 [math]\displaystyle{ \Gamma(n) = (n-1)! }[/math]

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:

[math]\displaystyle{ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,{\rm d}t. }[/math]

Particular values

Some particular values of the gamma function are:

[math]\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} }[/math]

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

[math]\displaystyle{ \Pi(z) = \Gamma(z+1) = z \; \Gamma(z) = \int_0^\infty e^{-t} t^{z+1}\,\frac{{\rm d}t}{t}, }[/math]

so that

[math]\displaystyle{ \Pi(n) = n! }[/math]

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:

[math]\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}. }[/math]

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")

[math]\displaystyle{ \zeta(z) \; \Gamma(z) = \int_0^\infty \frac{t^{z}}{e^t-1} \; \frac{dt}{t}. }[/math]

Gamma Function Media

• \Gamma(x+1) interpolates the factorial function to non-integer values.

• Gamma plus sin pi z.svg

  The gamma function, Γ(z) in blue, plotted along with Γ(z) + sin(πz) in green. Notice the intersection at positive integers. Both are valid analytic continuations of the factorials to the non-integers.

• Plot of gamma function in complex plane in 3D with color and legend and 1000 plot points created with Mathematica.svg

  Plot of the absolute value of the gamma function in complex plane in 3D with color and legend and 1000 plot points created with Mathematica

• Plot of gamma function in the complex plane from -2-i to 6+2i with colors created in Mathematica.svg

  Plot of gamma function in the complex plane from -2-2i to 6+2i with colors created in Mathematica

• Gamma cplot.svg

  Gamma function with domain coloring using cplot.

• Gamma abs 3D.png

  3-dimensional plot of the absolute value of the complex gamma function

• LogGamma Analytic Function.png

  The analytic function log Γ(z)

• Plot of logarithmic gamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

  Plot of logarithmic gamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

• Comparison gamma (blue line) with the factorial (blue dots) and Stirling's approximation (red line)

• 

  Daniel Bernoulli's letter to Christian Goldbach, October 6, 1729

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)
• 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).
• 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, archived from the original on 2021-10-27, retrieved 2013-02-10
• 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.