Complex-differentiable equals real differentiable plus Cauchy-Riemann differential equations

This article gives a proof/explanation of the equivalence of multiple definitions for the term function complex-differentiable at a point

View a complete list of pages giving proofs of equivalence of definitions

Statement

Verbal statement

The property of being complex-differentiable at a point is equivalent to the property of being real-differentiable at that point and satisfying the Cauchy-Riemann differential equations at that point.

Symbolic statement

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C}$ is a function. Let $z_{0}\in U$. Then $f$ is complex-differentiable at $z_{0}$ iff the following two equivalent conditions are satisfied:

• $f$ is real-differentiable at $z_{0}$
• $f$ satisfies the Cauchy-Riemann differential equations at $z_{0}$

Moreover the complex differential of $f$ at $z_{0}$ is the same complex number whose representation as a matrix is the real differential.