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This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this first volume, special emphasis is placed on geometric and complex variable methods involving integral representations. The following topics are treated: • integration and differentiation on manifolds • foundations of functional analysis • Brouwer's mapping degree • generalized analytic functions • potential theory and spherical harmonics • linear partial differential equations This new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy’s integral has been added. The second volume will present functional analytic methods and applications to problems in differential geometry. This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.

Differential equations, Partial. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Integral representations. --- Representations, Integral --- Partial differential equations --- Mathematics. --- Partial differential equations. --- Physics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Math --- Science --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Differential equations, partial. --- Mathematical physics. --- Physical mathematics --- Physics

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This comprehensive two-volume textbook presents the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is put on the connection of PDEs and complex variable methods. In this first volume the following topics are treated: Integration and differentiation on manifolds, Functional analytic foundations, Brouwer's degree of mapping, Generalized analytic functions, Potential theory and spherical harmonics, Linear partial differential equations. While we solve the partial differential equations via integral representations in this volume, we shall present functional analytic solution methods in the second volume. This textbook can be chosen for a course over several semesters on a medium level. Advanced readers may study each chapter independently from the others.

Differential equations, Partial --- Integral representations --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Partial differential equations --- Differential equations, partial. --- Mathematical physics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Mathematics --- Differential equations, Partial. --- Partial differential equations. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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Functional analysis --- Partial differential equations --- Mathematical physics --- differentiaalvergelijkingen --- functies (wiskunde) --- wiskunde --- fysica

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Functional analysis --- Partial differential equations --- Mathematical physics --- differentiaalvergelijkingen --- functies (wiskunde) --- wiskunde

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Das Buch bietet eine moderne Darstellung der Differential- und Integralrechnung für Funktionen in einer und mehreren reellen Veränderlichen sowie in einer komplexen Variablen. Die elementaren Funktionen werden über komplexe Potenzreihen definiert und die Logarithmusfunktion auf ihrer Riemannschen Fläche betrachtet. Nachdem die eindimensionale Integration mittels reeller und komplexer Stammfunktionen durchgeführt ist, wird über das uneigentliche n-dimensionale Riemannsche Integral die Integration auf Mannigfaltigkeiten mit Hilfe von Differentialformen vorgestellt. Mit dem Lebesgueschen Integral und dessen Maßtheorie werden die Banachräume p-fach integrierbarer Funktionen eingeführt. Es werden für gewöhnliche Differentialgleichungen systematisch Existenz-, Eindeutigkeits- und Stabilitätsfragen behandelt. In einem Kapitel zur Variationsrechnung wird direkt über die Untersuchung von Geodätischen der Riemannsche Raum und sein Krümmungsbegriff vorgestellt.

Mathematical analysis. --- Analysis (Mathematics). --- Differential equations. --- Differential geometry. --- Analysis. --- Ordinary Differential Equations. --- Differential Geometry.