Principal FunctionsDuring 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. |
Contents
| 1 | |
CHAPTER I | 38 |
4 SPECIAL TOPICS | 59 |
CHAPTER II | 81 |
2 CONFORMAL MAPPING | 87 |
3 REPRODUCING DIFFERENTIALS | 94 |
5 THE THEOREMS OF RIEMANNROCH AND ABEL | 114 |
6 EXTREMAL LENGTH | 121 |
2 OTHER PROPERTIES OF THE OCLASSES | 201 |
CHAPTER V | 211 |
CHAPTER VI | 232 |
and degeneracy 253 Other null classes 254 List of problems | 256 |
3 PRINCIPAL FORMS ON RIEMANNIAN SPACES | 272 |
Special cases | 285 |
APPENDIX | 305 |
2 MAXIMUM PRINCIPLE | 313 |
CHAPTER III | 138 |
2 EXTREMAL LENGTH | 150 |
3 EXPONENTIAL MAPPINGS OF PLANE REGIONS | 169 |
CHAPTER IV | 193 |
BIBLIOGRAPHY | 323 |
AUTHOR INDEX | 337 |
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Common terms and phrases
Ahlfors analytic arbitrary Riemann surface bordered boundary neighborhood bordered Riemann surface boundary component bounded circular slit compact bordered compact subset conformal mapping consider continuous contours convergence corresponding curves defined denote Dirichlet integral Dirichlet operator Dirichlet problem disjoint Existence Theorem extremal length function ƒ Green's formula H₁(A harmonic differential harmonic function harmonic measure harmonic p-form Hence holomorphic function homologous ideal boundary implies kernel L₁ Lemma logarithmic pole Main Existence Theorem mapping meromorphic function metric nonconstant normal operator obtain open Riemann surface open set order differential p-form p₁ parameter parametric disk partition planar plane region principal functions problem proof prove radial slit regular region regular subregion removable singularity reproducing differentials Riemann surface Riemannian spaces Sario satisfies singularity function slit annulus Suppose theory univalent zero