Prove: Let a curve C in the z plane, which may or may not be closed, be defined by the parametric equations
x = f(t)
y = g(t)
where f(t) and g(t) are assumed to be continuously differentiable. Then the transformation
1) z = f(w) + ig(w)
maps the real axis AB of the w plane onto curve C. See Fig. 1.
Proof. If z = x + iy and w = u + iv, the transformation can be written as
2) x + iy = f(u + iv) + ig(u + iv) .
The real axis in the w plane corresponds to v = 0. Setting v = 0 in 2) we get
x + iy = f(u) + ig(u)
i.e. x = f(u), y = g(u), which represents curve C.