The gamma function, denoted by , is deﬁned by the formula
which is defined for all complex numbers except the nonpositive integers. It is frequently used in identities and proofs in analytic contexts.
The above integral is also known as Euler's integral of second kind. It serves as an extension of the factorial function which is defined only for the positive integers. In fact, it is the analytic continuation of the factorial and is defined as
However, the gamma function is but one in a class of multiple functions which are also meromorphic with poles at the nonpositive integers.
The functional equation
is true for all values of ; this can be derived from an application of integration by parts.
Recalling the definition of the gamma function above, we can see that by applying integration by parts,
Hence, the functional equation holds.
Noting that , we can easily see for all positive integers by simple induction.
Using the functional equation for the gamma function, we obtain that
Consider the integral
Using integration by parts,
We can deduce that by integrating by parts times, we will get
Evaluating the integral, we have
Setting toward infintiy,
Then rewrite it as
The gamma function has a very nice duplication formula. It can be stated as
This is notoriously simple to prove.
We start with the representation of beta function and a relation between beta function and gamma function: Put and to get Change of variables and gives Now, using the property if we have Again using beta function, we have Since we have Using , we finally have Hence proved.
Euler's reflection formula is as follows:
Replace with to get
Multiply both of these to get
The product on the RHS is the famous Weierstrass product, so we have
Using Euler's reflection formula,
There is also an Euler reflection formula for the digamma function
Main article: Beta Function
The gamma and beta functions satisfy the identity
The above integral is indeed known as Euler's integral of first kind.
Main article: Digamma Function
The digamma function is defined as
Using this on the Euler reflection formula and Legendre duplication formula, we have
Main article: Riemann Zeta Function
The zeta function and gamma functions are very closely related.
The gamma function turns up in the zeta functional equations:
It also has an integral closely related to it:
Main article: Polylogarithm
An integral representation of gamma function is
The Bohr-Mollerup theorem states that the gamma function is the unique function on the interval such that and is a convex function.