Let for . The Gamma function determines how the integral of this function over the range varies with .
for . The Gamma function can be evaluated in R by the command gamma(a):
Properties:
Recurrence relation (see below): for .
Positive integers (see notes): , for a positive integer.
Special (see below): .
Limits (see below): as or .
Proofs:
Proof that for .
for .
Proof that .
In Math113 you saw that
Substituting , so , we have, by symmetry,
Proof that as .
as .
Proof that as .
as .