Integral formula

Definition

An integral formula or integral representation for a holomorphic function or meromorphic function $f:U\to \mathbb {C}$ is defined as a pair $(F,\gamma )$ where $F$ is a holomorphic function of two complex variables, and $\gamma$ is a curve in $\mathbb {C}$, such that we have the identity:

$f(z)=\int _{\gamma }F(z,w)\,dw$

on a nonempty open subset of $U$. In fact, equality holds wherever both sides make sense.

Examples

• Euler's integral formula for gamma function: This formula describes the gamma function in the right half-plane, even though the gamma function is actually defined on the complement of the set of non-positive integers.