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

The gamma function is defined for all complex numbers, but it is not defined for the non-positive integers. For a complex number whose real part is a positive integer, the function is defined by:^[2][3]

${\displaystyle \mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\mathrm{d}t.}$

The factorial definition of the gamma function can be derived using integration by parts.^[2] $${\displaystyle {\int }_a^{b}udv=[uv{]}_a^b-{\int }_a^{b}vdu.}$$ Let ${\displaystyle u=t^z}$ so ${\displaystyle du=z{t}^{z-1}}$ and ${\displaystyle dv=e^{-t}}$ so ${\displaystyle v=-e^{-t}}$ $${\displaystyle \mathrm{\Gamma }(z+1):={\int }_0^{\mathrm{\infty }}{t}^z{e}^{-t}dt}$$ so by applying integration by parts, $${\displaystyle \begin{array}{cc}\mathrm{\Gamma }(z+1)& =[-t^z{e}^{-t}{]}_0^{\mathrm{\infty }}+{\int }_0^{\mathrm{\infty }}z{t}^{z-1}{e}^{-t}dt=z\mathrm{\Gamma }(z).\end{array}}$$ as the first part vanishes (equals 0). Note that ${\displaystyle \mathrm{\Gamma }(1)=1}$. Thus, ${\displaystyle \mathrm{\Gamma }(z)=(z-1)!}$ for ${\displaystyle Re(z)>0}$, that is, real part of ${\displaystyle z}$ is positive.

Some particular values of the gamma function are:

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

Gauss introduced the Pi function. This is another way of denoting the gamma function. In terms of the gamma function, the Pi function is

${\displaystyle \mathrm{\Pi }(z)=\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{z+1}\frac{\mathrm{d}t}{t},}$

so that

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

for every non-negative integer n.

The gamma function is used to study the Riemann zeta function. A property of the Riemann zeta function is its functional equation:

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

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")

${\displaystyle \zeta (z)\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}\frac{t^z}{e^t-1}\frac{dt}{t}.}$

• Gamma distribution

• 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.