Pi function - Definition and Overview

In mathematics, the gamma function is a function that extends the concept of factorial to the complex numbers.

Definition

The notation Γ(z) is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral

<math>

\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt <math> converges absolutely. Using integration by parts, one can show that

<math>\Gamma(z+1)=z\Gamma(z)\,.<math>

Because Γ(1) = 1, this relation implies that

<math>\Gamma(n+1)=n\Gamma(n)=...=n!\Gamma(1)=n!\,<math>

for all natural numbers n. It can further be used to extend Γ(z) to a meromorphic function defined for all complex numbers z except z = 0,  −1, −2, −3, ... by analytic continuation.

It is this extended version that is commonly referred to as the gamma function.

Another important functional equation for the gamma function is Euler's reflection formula

<math>\Gamma(1-z)\Gamma(z) = {\pi \over \sin \pi z}.<math>

An alternative notation which is sometimes used is the Pi function, which in terms of the gamma function is

<math>\Pi(z) = \Gamma(z+1) = z\Gamma(z).<math>

We also sometimes find

<math>\pi(z) = {1 \over \Pi(z)}\,<math>

which is an entire function, defined for every complex number. That π(z) is entire entails it has no poles, so Γ(z) has no zeros.

Perhaps the most well-known value of the gamma function at a non-integer argument is

<math>\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.<math>

The gamma function has a pole of order 1 at z = −n for every natural number n; the residue there is given by

<math>\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}.<math>

The following infinite product for the gamma function, due to Weierstrass, is valid for all complex numbers z which are not non-positive integers:

<math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}<math>

where γ is the Euler-Mascheroni constant.

The Bohr-Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex.

Relation to other functions

In the first integral above, which defines the gamma function, the limits of integration are fixed. The incomplete gamma function is the function obtained by allowing either the upper or lower limit of integration to be variable.

The derivative of the logarithm of the gamma function is called the digamma function; higher derivatives are the polygamma function.

See also

• beta function
• Stirling's approximation
• multivariate gamma function

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6.); wiki: Abramowitz and Stegun.

• G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)

• W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 6.1.)

External links

• Examples of problems involving the Gamma function can be found at Exampleproblems.com (http://www.exampleproblems.com/wiki/index.php?title=Special_Functions).

• Gamma function at MathWorld (http://mathworld.wolfram.com/GammaFunction.html)