Suppose that
is a model flow velocity function with domain
The conjugate velocity function
is analytic on
may not be simple but if
is a simply connected subregion of
has has a primitive
satisfying 
for
A function
which is a primitive of a complex velocity functionis called a complex potential function for the flow. Taking the complex conjugate of (1) above gives
for
The circulation ofalong a curve
drawn in the fluid is
where the integral is evaluated alongand the flux ofacrossis
where the integral is evaluated along
Hence
where the integral is evaluated along
Provided that %GAMMA lies within the simply connected subregion S on which the complex potential %OMEGA is defined, we may apply the Fundamental Theorem of Algebra to deduce
where
and
are the start and end points of the curve
Hence
and
Hence the circulation or flux along any contour betweenanddepends only on the points %alpha and %beta and is independent of the contour.
If the flow at each point is tangential to a curvethen the flow acrossat each point ofis zero. Curves with this property are called streamlines and represent snapshots of the motion of the fluid. If a curve is a streamline then
for all pointsandonso that