TABLE OF CONTENTS This appendix serves as an introduction to the gamma and beta functions. These are special functions that find wide applications in science and engineering. They are also used in probability and in the computation of certain integrals.
The gamma function, denoted as ?( n), is also known as generalized factorial function. It is defined as
and this improper [*] integral converges (approaches a limit) for all n > 0.
We will derive the basic properties of the gamma function and its relation to the well known factorial function
We will evaluate the integral of (B.1) by performing integration by parts using the relation
Letting
we get
Then, with (B.3), we write (B.1) as
With the condition that n > 0, the first term on the right side of (B.6) vanishes at the lower limit, that is, for x = 0. It also vanishes at the upper limit as x ? ?. This can be proved with L' H pital's rule [*] by differentiating both numerator and denominator m times, where m ? n. Then,
Therefore, (B.6) reduces to
and with (B.1) we have
By comparing the two integrals of (B.9), we see that
or
It is convenient to use (B.10) for n < 0, and (B.11) for n > 0.
From (B.10), we see that ?( n) becomes infinite as n ? 0.
For n
