Let 
and 
on some region 
containing the point 
. If 
satisfies the Cauchy-Riemann
equations and has continuous first partial
derivatives in the neighborhood of 
, then 
exists and is given by
 |
and the function is said to be complex differentiable (or, 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.
