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,

and

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.