MathGroup Archive 2012

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Conformal Mapping

  • To: mathgroup at
  • Subject: [mg128541] Re: Conformal Mapping
  • From: Murray Eisenberg <murray at>
  • Date: Fri, 2 Nov 2012 00:43:26 -0400 (EDT)
  • Delivered-to:
  • Delivered-to:
  • Delivered-to:
  • Delivered-to:
  • References: <>

On Nov 1, 2012, at 3:19 AM, MaxJ <maxjasper at> wrote:
> I need help finding a Mobius transform such that the region:
> |z-i| < sqrt(2)
> &&
> |z+i| < sqrt(2)
> in z-plane be mapped conformally into a unit circle in w-plane.
> Any help is appreciated very much.

I assume you really did mean "and" rather than "or" in describing the region.

This seems more like a mathematics question than a Mathematica question.

Mathematica can help peripherally. The two circles bounding the region obviously intersect at complex points z = -1 and z = 1.  They intersect the imaginary axis at the points found from:

{I*(1 - Sqrt[2]), I*(-1 + Sqrt[2])}

And you may easy plot the region by using David Park's "Presentations" application, which allows you to express things directly in terms of complex numbers:

  << Presentations`

     ComplexRegionDraw[Abs[z - I] < Sqrt[2] && Abs[z + I] < Sqrt[2], {z, -2 - 2 I, 2 + 2 I},
                       BoundaryStyle -> Directive[Thick, Dashed]],
     PointSize[Large], ComplexPoint /@ pts
     Axes -> True]

As to the mathematics: the region is "lens-shaped". Consider what the Moebius transformation you seek does to the boundary -- surely maps it onto the unit circle. Consider the inverse of that transformation. Now apply the theorem that the image of a circle under any Moebius transformation is a circle (in the extended complex plane or, equivalently, on the Riemann sphere).

Murray Eisenberg                           murray at
Mathematics & Statistics Dept.      
Lederle Graduate Research Tower            phone 413 549-1020 (H)
University of Massachusetts                      413 545-2838 (W)
710 North Pleasant Street                  fax   413 545-1801
Amherst, MA 01003-9305

  • Prev by Date: Formatting Dynamic Output
  • Next by Date: Re: [newb] simple equation substitutions
  • Previous by thread: Conformal Mapping
  • Next by thread: Re: Conformal Mapping