Functions for Statistics

A form of integral which shows up regularly in quantum statistics is

The evaluation of this class of integrals is in terms of two special functions, the gamma function and the Riemann zeta function.

The Gamma Function

A function which shows up often in statistical calculations is the gamma function:

This can be related to another integral

leading to the relationships

Some other common relationships with gamma functions:

One of the applications of the gamma function is in the evaluation of statistical integrals.

The Riemann Zeta Function

A function which shows up often in statistical calculations is the Riemann zeta function:

One of the applications of the zeta function is in the evaluation of statistical integrals.