Overview

Whenever two or more objects or entities—be they bubbles, vortices, black holes, magnets, colloidal particles, microorganisms, swimming bacteria, Brownian random walkers, airfoils, turbine blades, electrified drops, magnetized particles, dislocations, cracks, or heterogeneities in an elastic solid—interact in some ambient medium, they make holes in that medium. Such holey regions with interacting entities are called multiply connected. This book describes a novel mathematical framework for solving problems in two-dimensional, multiply connected regions. The framework is built on a central theoretical concept: the prime function, whose significance for the applied sciences, especially for solving problems in multiply connected domains, has been missed until recent work by the author. This monograph is a one-of-a-kind treatise on the prime function associated with multiply connected domains and how to use it in applications. The book contains many results familiar in the simply connected, or single-entity, case that are generalized naturally to any number of entities, in many instances for the first time. Solving Problems in Multiply Connected Domains is aimed at applied and pure mathematicians, engineers, physicists, and other natural scientists; the framework it describes finds application in a diverse array of contexts. The book provides a rich source of project material for undergraduate and graduate courses in the applied sciences and could serve as a complement to standard texts on advanced calculus, potential theory, partial differential equations and complex analysis, and as a supplement to texts on applied mathematical methods in engineering and science.

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9781611976144

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Darren Crowdy has been a professor of applied mathematics at Imperial College London since 1999. An internationally recognized leader in applied and computational complex analysis, he has received many professional accolades, including a 2004 Philip Leverhulme Prize for his work on quadrature domains in ?uid mechanics and a 2009 CMFT Young Investigator Award for his work on the Schwarz–Christoffel problem in function theory. He is a Wolfson Royal Society Research Fellow, an EPSRC Established Career Fellow and Advanced Research Fellow, and an elected Fellow of the IMA. In 2008, his solution of a 140-year old problem in complex analysis and conformal geometry garnered extensive international media attention.

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