The gamma function

The gamma function is one of the classical functions of applied mathematics; here I will provide a bare bones introduction to it (see Connections for places to go learn more). You should think of it in the same way that you think about sin, cos, exp, and log. First, these functions have a specific mathematical definition. Second, there are known rules that relate functions with different arguments (such as the rule for computing sin(a + b)) and there are computational means for obtaining their values. Third, these functions are tabulated (in the old days, in tables of books, and in the modern days in many software packages or on the web).

The same applies to the gamma function, which is defined for z > 0 by r(z) =

In this expression, z can take any positive value, but let us start with the integers. In fact, let us start with z = 1, so that we consider r(1) = J0°° e-sds = 1. What about z = 2? In that case r(2) = J0°° se—sds, which can be integrated by parts and we find T(2) = 1. We shall do one more, before the general case: r(3) = J0°° s2e-sds, which can be integrated by parts once again and from which we will see that r(3) = 2. If you do a few more, you should get a sense of the pattern: for integer values of z, r(z) = (z — 1)!. Note, then, that we could write the binomial coefficient in Eq. (3.49) as n — 1 \ (n — 1)! r(n)

For non-integer values of z, the same kind of integration by parts approach works and leads us to an iterative equation for the gamma function, which is

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