Holomorphic function
A holomorphic function is a differentiable complex function. That is, just as in the real case,
is holomorphic at
if
exists. This is much stronger than in the real case since we must allow
to approach zero from any direction in the complex plane.
Cauchy-Riemann Equations
Let us break into its real and imaginary components by writing
, where
and
are real functions. Then it turns out that
is holomorphic at
iff
and
have continuous partial derivatives and the following equations hold:
These equations are known as the Cauchy-Riemann Equations.
Analytic Functions
A related notion to that of homolorphicity is that of analyticity. A function is said to be analytic at
if
has a convergent power series expansion on some neighborhood of
. Amazingly, it turns out that a function is holomorphic at
if and only if it is analytic at
.