A function 
on a region 
is complex differentiable at 
if 
exists and is given by
 |
where 
.
In particular, this derivative exists if 
satisfies the Cauchy-Riemann
equations and has continuous first partial
derivatives in the neighborhood of 
. Such a function is equivalently analytic
or holomorphic.
A function 
can be thought of as a map from the plane to the plane, 
. Then 
is complex differentiable iff its
Jacobian is of the form
![[a -b; b a]](/images/equations/ComplexDifferentiable/NumberedEquation2.svg) |
at every point. That is, its derivative is given by the multiplication of a complex number 
.
For instance, the function 
, where 
is the complex conjugate,
is not complex differentiable.
