If
is a continuous complex velocity function on a region
and suppose that
and 
have partial derivatives with respect to
and
which are continuous on
Then is a model flow velocity function onif and only if
and
Proof
By the Cauchy – Riemann equations, if the partial derivatives of the real functions
and
exist and are continuous on a region
then
is analytic onif and only if
and
on
The velocity function
has conjugate velocity function
andis a model flow velocity function onif and only if
is analytic onPutting
and
we have thatis a model flow velocity function onif and only if
(1) and
(2) on R.
Differentiating (1) with respect togives
and differentiating (2) with respect togives
Subtracting these equations and using that
for analytic functions gives
(3)
A similar equation can be derived for
Equation (30 is Laplace's equation.