Cauchy-Riemann Equations

Suppose is Holomorphic and
For a fixed , take the derivative in direction:

Similarly, for a fixed , take derivative in direction:

Equating we find the Cauchy-Riemann equations:


These are not sufficient conditions for to be differentiable.

Theorem

Suppose .
Then is differentiable at
if and only if
and are differentiable at
and their partial derivatives satisfy and .

Corollary

is differentiable at
if and only if
partial derivatives of and exist and are continuous around
and they satisfy the Cauchy-Riemann equations.
Proved in Analysis and Topology.