Gamma function

The Gamma function is an important function that is useful in many areas of applied mathematics, particularly in complex analysis. It can be thought of as a generalization of the standard high-school factorial to the complex numbers.

The Gamma function occurs in statistics since is closely related to the Gamma distribution. It is one of the simpler members of the so-called special functions which appear throughout mathematics. Consequently, it provides a good foundation for study of the Beta function, the Riemann zeta function, and the hypergeometric functions.

A common mathematical practice is to leave an answer to a problem in terms of the Gamma function, in the same way that one might leave an answer in terms of a sine or cosine. Also, a good working knowledge of the Gamma function can speed up many calculations in mathematics. For example, knowing the value of Γ(1/2) shown below can speed up the evaluation of many integrals that occur in the study of the Gaussian distribution.

Definition

The Gamma function is a meromorphic function on the whole of the complex plane. It is defined as

Γ(z) = integral(exp(-t)tz-1dt, t=0...∞)

for Re(z)>0, where we take the principal value of t^z-1. If Re(z) is negative then the integral is undefined, since the t^z-1 factor cannot be integrated near 0.

We use the mathematical idea of analytic continuation to extend the Gamma function to the rest of the complex numbers. Using mathematical analysis, we can prove Γ(z) is analytic, and consequently there is a unique way to extend it to left half plane while retaining analyticity. A brute force method of doing this would be using Taylor expansions, but instead we use the much more elegant approach which arises from the basic properties below.

Basic properties

For natural numbers the Gamma function can be evaluated directly. For example

Γ(1) = integral(exp(-t)t1-1dt, t=0...∞)
     = integral(exp(-t)dt, t=0...∞)
     = 1.

The most important property of the Gamma function is

Γ(z+1) = zΓ(z).

This can be seen from the definition using integration by parts. However, because the integral is infinite, a full mathematical justification requires some analysis about limit switching. One immediate consequence of this is that for any natural number n,

Γ(n+1) = n Γ(n) = n(n-1)...(2)(1) Γ(1) = n!.

The identity can also be used to find the analytic continuation of Γ to the left half plane. Suppose we consider the function f(z) = Γ(z+1)/z, which is defined on Re(z)>-1. Then by the property above, we see that f(z) - Γ(z) is identically zero in Re(z)>0. Hence f(z) must be the analytic continuation of Γ(z) to Re(z)>-1. Similarly, we may define Γ(z) for Re(z)>-n as

Γ(z) = Γ(z+n) /((z+n-1)(z+n-2)...(z))

This expression shows us that Γ has simple poles at all the non-positive integers.

The reflection formula

The reflection formula states that

Γ(z)Γ(1-z) = πcosec(πz)

and can be proved using the properties of the Beta function. This turns out to be a very useful identity. It gives us an alternative way of finding the analytic continuation of Γ to the left half plane. It also allows us to find Γ(1/2):

Γ(1/2)Γ(1-1/2) = πcosec(π/2)
     (Γ(1/2))² = π
        Γ(1/2) = sqrt(π).

The Hankel representation

The Hankel representation is another integral representation for Γ. Although it is more complicated than the standard definition, it is defined over the whole of the complex plane, so we don't need any analytic continuation. We have

Γ(z) = 1/(2isin(πz)) integral(exp(t)tz-1dt, C)

Where C is the Hankel contour in the complex plane, which circles the origin as shown in the diagram.

          |Im
          |
  ---<----+\
          | \      Re
  --------0--+-------
          | /
  --->----+/
          |
          |

Using the the reflection formula, we can find a corresponding representation for 1/Γ as follows

1/Γ(z) = Γ(1-z)/(πcosec(πz))
       = 1/(2πi) integral(exp(t)tz-1dt, C)

Since the integral is defined for all z in the complex plane, we deduce that 1/Γ has no singularities, and from this we can see that Γ(z) has no zeros.

The Gamma function or the Factorial function is just what its name says. An extension of the factorial to real numbers. Thust the basic property that we would require from this function is that it match with the factorial for positive integers. By some strange convention the gamma function(I'll use g here instead of gamma) is defined so that g(n)=(n-1)! . The second major property that we expect from a factorial is a recursion relation of the kind (n+1)! = (n+1)*n!. Because of the above convention the gamma function must satisfy g(x+1)=xg(x)

There are a number of ways of defining the gamma function, all equivalent though. The standard modern definition is to define it as
g(x) = integral from 0 to inf (e^-t t^x-1)
This improper integral converges for x>0.
Integrating this by parts, its easy to show that g(x) satisifies the above recursion relation. Then since its obvious that g(1)=1, the recursion relation allows us to show that g(n)=(n-1)! for integer n.
An interesting property that g has is that
g(1/2) = sqrt(pi)

The initial definition of g by Euler was in the form of an infinite product,but that is not very useful except perhaps for proving the relation
g(x)g(1-x)=pi/sin(pi*x)

Various substitutions into the above integral yield different forms of the definition. For negative values of x the above improper integral does not converge, so we need to look at an analytic continuation. The easiest way to define this is using the recursion relation. Since we have g(x-1) = (g(x))/x-1 we can use the values of g in (0,1) for the values of g in (-1,0). This process may now be extended indefinitely. For negative integers and 0 it is true that
lim(x->-n) 1/g(x) = 0.

The gamma function is useful for evaluating various definite integrals involving exponents and is related to the beta function by means of
b(x,y)=g(x)*g(y)/(g(x+y))