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B. Rodin, L. Sario

Principal Functions

Paperback Engels 2012 9781468480405

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Samenvatting

During the decade and a half that has elapsed since the intro duction of principal functions (Sario [8 J), they have become impor tant tools in an increasing number of branches of modern mathe matics. The purpose of the present research monograph is to systematically develop the theory of these functions and their ap plications on Riemann surfaces and Riemannian spaces. Apart from brief background information (see below), nothing contained in this monograph has previously appeared in any other book. The basic idea of principal functions is simple: Given a Riemann surface or a Riemannian space R, a neighborhood A of its ideal boundary, and a harmonic function s on A, the principal function problem consists in constructing a harmonic function p on all of R which imitates the behavior of s in A. Here A need not be connected, but may include neighborhoods of isolated points deleted from R. Thus we are dealing with the general problem of constructing harmonic functions with given singularities and a prescribed behavior near the ideal boundary. The function p is called the principal function corresponding to the given A, s, and the mode of imitation of s by p. The significance of principal functions is in their versatility.

Specificaties

ISBN13:9781468480405

Taal:Engels

Bindwijze:paperback

Aantal pagina's:348

Uitgever:Springer New York

Serie:The university series in higher mathematics

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Inhoudsopgave

Introduction: What are Principal Functions?.- 0 Prerequisite Riemann Surface Theory.- §1. Topology of Riemann Surfaces.- §2. Analysis on Riemann Surfaces.- I The Normal Operator Method.- §1. The Main Existence Theorem.- §2. Normal Operators.- §3. The Principal Functions p0 and p1.- §4. Special Topics.- II Principal Functions.- §1. Main Extremal Theorem.- §2. Conformal Mapping.- §3. Reproducing Differentials.- §4. Interpolation Problems.- §5. The Theorems of Riemann-Roch and Abel.- §6. Extremal Length.- III Capacity Stability and Extremal Length.- §1. Generalized Capacity Functions.- §2. Extremal Length.- §3. Exponential Mappings of Plane Regions.- §4. Stability.- IV Classification Theory.- §1. Inclusion Relations.- §2. Other Properties of the O-Classes.- V Analytic Mappings.- §1. The Proximity Function.- §2. Analytic Mappings.- §3. Meromorphic Functions.- VI Principal Forms and Fields on Riemannian Spaces.- §1. Principal Functions on Riemannian Spaces.- §2. Principal Forms on Locally Flat spaces.- §3. Principal Forms on Riemannian Spaces.- VII Principal Functions on Harmonic Spaces.- §1. Harmonic Spaces.- §2. Harmonic Functions with General Singularities.- §3. General Principal Function Problem.- Appendix Sario Potentials on Riemann Surfaces.- §1. Continuity Principle.- §2. Maximum Principle.- Author Index.

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