simple example of composed conformal mapping

Let’s consider the mapping


where a and b are complex and  a0.

Because  f(z)a0,  the mapping is conformal in the whole z-plane.  Denote  a:=ϱeiα (where  ϱ,α) and

z1:=ϱz, (1)
z2:=eiαz1, (2)
w:=z2+b. (3)

Then the mapping  zz1  means a dilation in the complex plane, the mapping  z1z2  a rotationMathworldPlanetmath by the angle α and the mapping  z2w  a translation determined by the vector from the origin to the point b.  Thus f is composed of these three consecutive mappings which all are conformal.

Figure 1: The mapping from z to z1
Figure 2: The mapping from z1 to z2
Figure 3: The mapping from z2 to w
Title simple example of composed conformal mapping
Canonical name SimpleExampleOfComposedConformalMapping
Date of creation 2013-03-22 16:47:25
Last modified on 2013-03-22 16:47:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 30E20
Classification msc 53A30