Up to this point, we do not have any particular convenience in representing the flow in the complex plane. The full potential of this choice will become clear as soon as we introduce conformal mapping techniques. Let
be an analytic function. It follows that also the inverse function z(z') is analytic.
Consider the two planes z and
The above function creates a link between a point in the z plane and a point in the z' plane. We can state that it maps one plane to the other. This transformation is said to be conformal because it does not affect angles, in the sense that given two lines in the z plane that intersect with some angle, the two transformed lines in the z' plane intersect with the same angle. In particular, two orthogonal families of curves in the z plane map into two other orthogonal families of curves in the z' plane. It follows that a conformal transformation maps equipotential andstream lines of an irrotational flow in the z plane into the corresponding lines of another irrotational flow in the z' plane.
Given a flow field in the z plane with complex potential W(z), the function
is analytic because both W(z) and z(z') are analytic. In other words, the derivative
exists and is unique because the derivatives on the right hand side exist and are unique.
Therefore, W' is the complex potential of an irrotational inviscid flow in the z' plane.
If P and P' are two corresponding points in the z and z' planes, respectively,
so that the two complex potentials W and W' assume the same value in corresponding points of the two domains.
Circulation along any (corresponding) closed line has also the same value in the two spaces because it is given by the integrals
that are equal because along the two lines C and C', the potentials assume the same value.
Among the conformal transformations, the Joukowski transformation is relevant for the study of flow around a wing, because it maps the domain around a cylinder into the domain around a wing, whose thickness and curvature can be varied.