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 tz-1. If Re(z) is negative then the integral is undefined, since the tz-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.