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.