The harmonic conjugate to a given function 
is a function 
such that
 |
is complex differentiable (i.e., satisfies the Cauchy-Riemann equations). It is given by
 |
where 
,
,
and 
is a constant of integration.
Note that 
is a closed form since 
is harmonic, 
. The line integral
is well-defined on a simply
connected domain because it is closed. However, on a domain which is not simply
connected (such as the punctured disk), the harmonic conjugate may not exist.
