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 a 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
.