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# 11: Conformal Transformations

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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 specific family of conformal maps called fractional linear transformations and, in particular at their geometric properties. As an application we will use fractional linear transformations to solve the Dirichlet problem for harmonic functions on the unit disk with specified values on the unit circle. At the end we will return to some questions of fluid flow.

Thumbnail: A rectangular grid under a conformal map. It is seen that maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. (Public Domain; Oleg Alexandrov via Wikipedia)

11: Conformal Transformations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.