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# 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 $\Gamma(n) = (n-1)!$[1][2]

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]

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,{\rm d}t.$

## 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 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")

$\zeta(z) \; \Gamma(z) = \int_0^\infty \frac{t^{z}}{e^t-1} \; \frac{dt}{t}.$

## References

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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).
• Birkhoff, George D. (1913). "Note on the gamma function". Bull. Amer. Math. Soc. 20 (1): 1–10.
• P. E. Böhmer, ´´Differenzengleichungen und bestimmte Integrale´´, Köhler Verlag, Leipzig, 1939.
• Bonnar, James (2010). The Gamma Function. Createspace Independent Publishing.
      .

• 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,
      , http://apps.nrbook.com/empanel/index.html?pg=256

• 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.
      .

• Whittaker, E.T.; Watson, G.N. (1996). A Course of Modern Analysis. Cambridge University Press.
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