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The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.

Complex analysis --- Mathematical analysis --- Functions of complex variables --- Functions of complex variables. --- Mathematical analysis. --- Engineering & Applied Sciences --- Applied Mathematics --- Analysis (Mathematics). --- Analysis. --- 517.1 Mathematical analysis

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Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions. The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.) on parameters. The articles collected here can be grouped under three headings. (A) Singularities of real maps; (B) Singular complex variables; and (C) Singularities of homomorphic maps.

Singularities (Mathematics) --- 512.76 --- Geometry, Algebraic --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc. --- Singularités (Mathématiques) --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Algebraic geometry. --- Complex manifolds. --- Functions of real variables. --- Several Complex Variables and Analytic Spaces. --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Real Functions. --- Geometry, Differential --- Topology --- Real variables --- Functions of complex variables --- Analytic spaces --- Manifolds (Mathematics) --- Algebraic geometry --- Geometry --- Analysis, Global (Mathematics) --- Differential topology --- Complex variables --- Elliptic functions --- Functions of real variables

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Differential equations --- 517.91 --- Ordinary differential equations: general theory --- Boundary value problems. --- Differential equations. --- 517.91 Ordinary differential equations: general theory --- Boundary value problems --- 517.91 Differential equations --- Boundary conditions (Differential equations) --- Functions of complex variables --- Mathematical physics --- Initial value problems --- 517.91. --- Numerical solutions

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Differential equations --- Boundary value problems --- Differential equations, Elliptic --- -Riemannian manifolds --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- Boundary conditions (Differential equations) --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Numerical solutions --- Riemannian manifolds. --- Boundary value problems. --- Riemann, Variétés de. --- Problèmes aux limites. --- Équations différentielles elliptiques --- Numerical solutions. --- Solutions numériques. --- Riemannian manifolds --- Solutions numériques

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Riemannian, symplectic and complex geometry are often studied by means ofsolutions to systems ofnonlinear differential equations, such as the equa­ tions of geodesics, minimal surfaces, pseudoholomorphic curves and Yang­ Mills connections. For studying such equations, a new unified technology has been developed, involving analysis on infinite-dimensional manifolds. A striking applications of the new technology is Donaldson's theory of "anti-self-dual" connections on SU(2)-bundles over four-manifolds, which applies the Yang-Mills equations from mathematical physics to shed light on the relationship between the classification of topological and smooth four-manifolds. This reverses the expected direction of application from topology to differential equations to mathematical physics. Even though the Yang-Mills equations are only mildly nonlinear, a prodigious amount of nonlinear analysis is necessary to fully understand the properties of the space of solutions. . At our present state of knowledge, understanding smooth structures on topological four-manifolds seems to require nonlinear as opposed to linear PDE's. It is therefore quite surprising that there is a set of PDE's which are even less nonlinear than the Yang-Mills equation, but can yield many of the most important results from Donaldson's theory. These are the Seiberg-Witte~ equations. These lecture notes stem from a graduate course given at the University of California in Santa Barbara during the spring quarter of 1995. The objective was to make the Seiberg-Witten approach to Donaldson theory accessible to second-year graduate students who had already taken basic courses in differential geometry and algebraic topology.

Global analysis (Mathematics) --- Four-manifolds (Topology) --- Mathematical Theory --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Analyse globale (Mathematiques) --- Globale analyse (Wiskunde) --- Trois-variétés (Topologie) --- Vier-menigvuldigheden (Topologie) --- Analyse globale (Mathématiques) --- Variétés topologiques à 4 dimensions --- Algebra. --- Algebraic topology. --- Calculus of variations. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- System theory. --- Algebraic geometry. --- Algebraic Topology. --- Calculus of Variations and Optimal Control; Optimization. --- Global Analysis and Analysis on Manifolds. --- Systems Theory, Control. --- Algebraic Geometry. --- Algebraic geometry --- Systems, Theory of --- Systems science --- Science --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Mathematical analysis --- Philosophy