A Guide to Complex Variables
This quick and easy-to-use guide provides a solid grounding in the fundamental area of complex variables. Copious figures and examples are used to illustrate the principal ideas, and the exposition is lively and inviting. In addition to important ideas from the Cauchy theory, the author also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping, and dozens of other central topics. An undergraduate taking a first look at the subject, or a graduate student preparing for their qualifying exams, will find this book to be both a valuable resource and a useful companion to more exhaustive texts in the field. For mathematicians and non-mathematicians alike.
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The Complex Plane
Complex Line Integrals
Applications of the Cauchy Theory
The Argument Principle
The Geometric Theory
accumulation point analytic continuation argument principle boundary calculation Cauchy-Riemann equations circle closed curve compact sets compact subsets complex derivative complex line integral complex number conformal mapping conformal self-map connected open set constant continuous function converges normally converges uniformly dD(P defined definition deformed direct analytic continuation Dirichlet problem disc D(P entire function essential singularity example Figure finite func function element function f functions Let Fundamental Theorem global analytic function harmonic function holomorphic function homotopic infinite product isolated singularity Laurent expansion Laurent series linear fractional transformation Mathematical maximum principle mean value property meromorphic function modulus morphic function multiplicity neighborhood one-to-one open set plane polynomial power series principle for holomorphic proof re'e real axis real number removable singularity residue result Riemann surface root Ross Honsberger Schwarz reflection series expansion simply connected subharmonic functions Suppose theorem Let tion uniformly Cauchy uniformly on compact unit disc zero set