Complex differentiability of a function 
on a region 
at 
means that 
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 
. Complex differentiability at every point
of an open set is equivalent to being analytic
or holomorphic there. Complex differentiability
at a single point alone does not imply either property
in a neighborhood.
A function 
can be thought of as a map
from the plane to the plane,
.
At a point where this real map
is differentiable, 
is complex differentiable iff its
Jacobian is of the form
![[a -b; b a]](/images/equations/ComplexDifferentiable/NumberedEquation2.svg) |
at that point. That is, its derivative is given by multiplication by a complex
number 
.
For instance, the function 
, where 
is the complex conjugate,
is not complex differentiable.
