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 [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

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

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  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.

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  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

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

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  Gamma function with domain coloring using cplot.

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

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  The analytic function log Γ(z)

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

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

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  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).
• Birkhoff, George D. (1913). "Note on the gamma function". Bull. Amer. Math. Soc. 20 (1): 1–10. doi:10.1090/S0002-9904-1913-02429-7. MR 1559418. S2CID 120440543.
• 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.