Complex Derivative

Let be a function where
We say that is complex differentiable at
if there is some constant and we can write:

as (i.e. )
We say that is the complex derivative of at
We can check that is unique (if it exists)
We can then write:

Example

The function is complex differentiable at
with complex derivative
It is not differentiable anywhere else.

Proof

Let and
Then and suppose derivative exists
Then:

Try first and :

Thus we find ought to be if it exists
Try now and :

So ought to be if it exists.
So we find