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|Title:||Exterior Riemann Map|
|Abstract:||Conformal maps are a way of distorting regions in the plane in such a manner that angles between intersecting curves are preserved. Owing to their connection to partial differential equations, conformal maps have a variety of uses in science and engineering. The Riemann mapping theorem guarantees the existence of a conformal map between two simply connected regions in the plane. A consequence is that the exteriors of such regions can also be mapped onto each other conformally. This picture shows how the exterior of the circle gets mapped conformally onto the exterior of the star-shaped region. The loops surrounding the star shape were originally concentric circles that have been transformed by the mapping. Notice that the angles between the loops and the outgoing rays are perpendicular, which shows the angle preservation property. The rainbow color gradient indicates which points on the circle correspond to which points on the star shape. My research is on integral equation methods for solving partial differential equations. While analyzing a method I was studying, I found an unexpected connection between the solution of a certain integral equation and the exterior Riemann map. This image was generated using the computed solution.|
|Rights Information:||Copyright 2016 Matt Wala|
|Date Available in IDEALS:||2016-04-15|