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AuthorRodin, Burton. author
TitlePrincipal Functions [electronic resource] / by Burton Rodin, Leo Sario
ImprintNew York, NY : Springer New York : Imprint: Springer, 1968
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Descript XVIII, 348 p. 1 illus. online resource


During the decade and a half that has elapsed since the introยญ duction of principal functions (Sario [8 J), they have become imporยญ tant tools in an increasing number of branches of modern matheยญ matics. The purpose of the present research monograph is to systematically develop the theory of these functions and their apยญ 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


Introduction: What are Principal Functions? -- 0 Prerequisite Riemann Surface Theory -- ยง1. Topology of Riemann Surfaces -- ยง2. Analysis on Riemann Surfaces -- I The Normal Operator Method -- ยง1. The Main Existence Theorem -- ยง2. Normal Operators -- ยง3. The Principal Functions p0 and p1 -- ยง4. Special Topics -- II Principal Functions -- ยง1. Main Extremal Theorem -- ยง2. Conformal Mapping -- ยง3. Reproducing Differentials -- ยง4. Interpolation Problems -- ยง5. The Theorems of Riemann-Roch and Abel -- ยง6. Extremal Length -- III Capacity Stability and Extremal Length -- ยง1. Generalized Capacity Functions -- ยง2. Extremal Length -- ยง3. Exponential Mappings of Plane Regions -- ยง4. Stability -- IV Classification Theory -- ยง1. Inclusion Relations -- ยง2. Other Properties of the O-Classes -- V Analytic Mappings -- ยง1. The Proximity Function -- ยง2. Analytic Mappings -- ยง3. Meromorphic Functions -- VI Principal Forms and Fields on Riemannian Spaces -- ยง1. Principal Functions on Riemannian Spaces -- ยง2. Principal Forms on Locally Flat spaces -- ยง3. Principal Forms on Riemannian Spaces -- VII Principal Functions on Harmonic Spaces -- ยง1. Harmonic Spaces -- ยง2. Harmonic Functions with General Singularities -- ยง3. General Principal Function Problem -- Appendix Sario Potentials on Riemann Surfaces -- ยง1. Continuity Principle -- ยง2. Maximum Principle -- Author Index

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