Gamma function

Last week I was teaching about combinatorial analysis and talking about factorial operator I remembered Gamma Function, this function is very useful in signal processing but particularly Gamma Function is equal to the factorial for non-negative integer numbers.

Theory about Gamma Function is very well described in Wikipedia: http://en.wikipedia.org/wiki/Gamma_function

The equation that defines Gamma Function is

Following figure presents the graph of Gamma Function

And when z is a non-negative integer is verified that

This consequence is because of the property

Typical values of Gamma function are

 In Scilab, there are both functions: gamma(.) and factorial(.), following are some examples of these functions


-->factorial(1)
 ans  =

    1. 

-->gamma(1)
 ans  =

    1. 

-->factorial(1.5)
 !--error 10000
factorial: Wrong value for input argument #1: Scalar/vector/matrix/hypermatrix of positive integers expected.
at line      14 of function factorial called by : 
factorial(1.5)


-->gamma(1.5)
 ans  =

    0.8862269 

-->sqrt(%pi)/2
 ans  =

    0.8862269 

Look factorial(.) is not possible to be applied in a non-integer number, the same happens with negative numbers.

And gamma(.) in 1.5 is equal to sqrt(%pi)/2 verifying the correspondence presented in the typical values figure.


Obs.: all equations and figures that I posted in this text were got from the Wikipedia page.