Topic 10 Notes Jeremy Orlo - MIT Mathematics

Topic 10 NotesJeremy Orloff10 conformal IntroductionIn this Topic we will look at the geometric notion of conformal maps. It will turn outthat analytic functions are automatically conformal . Once we have understood the generalnotion, we will look at a specific family of conformal maps called fractional linear trans-formations and, in particular at their geometric properties. As an application we will usefractional linear transformations to solve the Dirichlet problem for harmonic functions onthe unit disk with specified values on the unit circle. At the end we will return to somequestions of fluid Geometric definition of conformal mappingsWe start with a somewhat hand-wavy definition:Informal maps are functions onCthat preserve the angles precisely: Supposef(z) is differentiable atz0and (t) is a smooth curve be concrete, let s suppose (t0)

Jeremy Orlo 10 Conformal transformations 10.1 Introduction In this topic we will look at the geometric notion of conformal maps. It will turn out that analytic functions are automatically conformal. Once we have understood the general notion, we will look at a speci c family of conformal maps called fractional linear trans-

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