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.
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.
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 .