Many important functions in applied sciences are defined via improper
integrals. Maybe the most famous among them is the Gamma
Function. This is why we thought it would be a good idea to have a
page on this function with its basic properties. You may consult any
library for more information on this function.
Historically the search for a function generalizing the factorial expression for the natural numbers was on. In dealing with this problem one will come upon the well-known formula
A very quick approach to this problem suggests to replace n by x in the improper integral to generate the function
Clearly this definition requires a close look in order to determine the domain of f(x). The only possible bad points are 0 and . Let us look at the point 0. Since when , then we have
when . The p-test implies that we have convergence around 0 if and only if -x < 1 (or equivalently x >-1). On the other hand, it is quite easy to show that the improper integral is convergent at regardless of the value of x. So the domain of f(x) is . If we like to have as a domain, we will need to translate the x-axis to get the new function
which explains somehow the
awkward term x-1 in the power of t. Now the domain of
this new function (called the Gamma Function) is
. The above formula is also known as Euler's second integral
(if you wonder about Euler's first integral, it is coming a little
later).
Basic Properties of
for any x > 0. In order to show this formula from the definition of , we will use the following identity
(this is just an integration by parts). If we let a goe to 0 and
b goe to , we get the desired identity.
In particular, we get
for any x > 0 and any integer . This formula makes it possible for the function to be extended to (except for the negative integers). In particular, it is enough to know on the interval (0,1] to know the function for any x > 0. Note that since
we get . Combined with the above identity, we get what we expected before :
or equivalently
for every and x >0. If we let n goe to , we obtain the identity
Note that this formula identifies the Gamma function in a unique fashion.
Knowing that the sequence
converges to the constant -C, where
If we take the derivative we get
or
In fact, one can differentiate the Gamma function infinitely often. In "analysis" language we say that is of -class. Below you will find the graph of the Gamma function.
It is defined for two variables x and y. This is an improper integral of Type I, where the potential bad points are 0 and 1. First we split the integral and write
When , we have
and when , we have
So we have convergence if and only if x > 0 and y >0 (this is done via the p-test). Therefore the domain of B(x,y) is x > 0 and y>0. Note that we have
Let a and b such that , we have (via an integration by parts)
If we let a goe to 0 and b goe to 1, we will get
Using the properties of the Gamma function, we get
or
In particular, if we let x=y = 1/2, we get
If we set or equivalently , then the technique of substitution implies
Hence we have
or
Using this formula, we can now easily calculate the value of .
where
If we take, x=n, we get after multiplying by n
This is a well known result, called Stirling's formula. So for large n, we have
which implies
Using the Weierstrass product formula (for and ), we get
If we use the Beta function (B(x,y)), we get the following formulas:
This page is inspired by Emil Artin's book on the Gamma Function. The exact reference is: Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964.
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.
Mohamed A. Khamsi