The Gamma function is a generalization of the factorial function to non-integer numbers.
Recall that, if , its factorial isso that satisfies the following recursion:
The Gamma function satisfies a similar recursion:but it is defined also when is not an integer.
The following is a possible definition of the Gamma function.
Definition The Gamma function is a function satisfying the following equation:
While the domain of definition of the Gamma function can be extended beyond the set of strictly positive real numbers (for example to complex numbers), the somewhat restrictive definition given above is more than sufficient to address all the problems involving the Gamma function that are found in these lectures.
Given the above definition, it is straightforward to prove that the Gamma function satisfies the following recursion:
The recursion can be derived by using integration by parts:
When the argument of the Gamma function is a natural number then its value is equal to the factorial of :
First of all, we have that
Using the recursion , we obtain
The following sections contain more details about the Gamma function.
A well-known fact, which is often used in probability theory and statistics is the following:
By using the definition and performing a change of variable, we obtain
By using this fact and the recursion formula previously shown, it is immediate to prove thatfor .
The result is obtained by iterating the recursion formula:
There are also other special cases in which the value of the Gamma function
can be derived analytically, but it is not possible to express
in terms of elementary functions for every
As a consequence, one often needs to resort to numerical algorithms to compute
For example, the Matlab command
value of the Gamma function at the point
For a thorough discussion of a number of algorithms that can be employed to compute numerical approximations of see Abramowitz and Stegun (1965).
The definition of the Gamma function:can be generalized by substituting the upper bound of integration () with a variable ():The function thus obtained is called lower incomplete Gamma function.
Below you can find some exercises with explained solutions.
Compute the following ratio:
We need to repeatedly apply the recursive formulato the numerator of the ratio:
We need to use the relation of the Gamma function to the factorial function: which, for , becomes
Express the following integral in terms of the Gamma function:
This is accomplished as follows:where in the last step we have just used the definition of Gamma function.
Abramowitz, M. and I. A. Stegun (1965) Hanbook of mathematical functions: with formulas, graphs, and mathematical tables, Courier Dover Publications.
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