[ Home ] [ Up ] [ Info ] [ Mail ]

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.

[ Home ] [ Up ] [ Info ] [ Mail ]