# The kernel function and conformal mapping

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*S Bergman*

The

The unification of methods in the theory of functions of one and several
complex variables is one of the purposes of introducing the kernel function and
the domains with a distinguished boundary. This approach has been extensively
developed during the last two decades. This second edition of Professor
Bergman's book reviews this branch of the theory including recent developments
not dealt with in the first edition. The presentation of the topics is simple
and presupposes only knowledge of an elementary course in the theory of
analytic functions of one variable.

#### Table of Contents

# Table of Contents

## The kernel function and conformal mapping

- CONTENTS ix10 free
- I. ORTHOGONAL FUNCTIONS 112 free
- II. THE KERNEL FUNCTION AND ASSOCIATED MINIMUM PROBLEMS 2132
- III. THE INVARIANT METRIC AND THE METHOD OF THE MINIMUM INTEGRAL 3142
- IV. KERNEL FUNCTIONS AND HILBERT SPACE 4253
- V. REPRESENTATION OF THE CLASSICAL DOMAIN FUNCTIONS 4657
- 1. The Dirichlet integral and the classical functions 4657
- 2. Orthogonal harmonic functions 4960
- 3. Representation of Green's and Neumann's functions 5263
- 4. Further identities involving the kernel function 5970
- 5. Some further domain functions 6273
- 6. The l-transforms 6778
- 7. The eigenfunctions of the l-kernel 7182
- 8. Discussion of the eigenfunctions 7586
- 9. The space Λ[sub(s)] and its kernel functions 8091
- 10. Applications to the theory of univalent functions 8697

- VI. CANONICAL CONFORMAL TRANSFORMATIONS 89100
- VII. ORTHOGONALIZATION OVER THE BOUNDARY 108119
- VIII. VARIATIONAL METHODS 122133
- IX. EXISTENCE PROOFS 138149
- X. PARTIAL DIFFERENTIAL EQUATIONS 144155
- XI. FUNCTIONS OF TWO COMPLEX VARIABLES AND PSEUDO-CONFORMAL MAPPINGS 166177
- 1. Examples of domains and their visualization 166177
- 2. Geometry of the space of two complex variables 170181
- 3. Orthogonal functions of two complex variables 176187
- 4. A metric invariant with respect to PCT's 182193
- 5. Representative domains 187198
- 6. Interior distinguished sets 194205
- 7. The method of minimum integrals 198209
- 8. The behavior of the kernel function K[sub(B)](z,z) in the neighborhood of the boundary 203214
- 9. Further bounds for the length of the line element of the invariant metric 206217

- XII. GENERALIZATION OF POTENTIAL-THEORETICAL AND CERTAIN SUBCLASSES OF FUNCTIONS 211222
- 1. Certain subclasses of functions f(z[sub(1)],z[sub(2)]) in domain with a distinguished boundary surface 211222
- 2. Functions of the extended class 214225
- 3. A generalized Poisson-Jensen formula for analytic polyhedra 216227
- 4. The generalized Nevanlinna formula 219230
- 5. Inequalities for certain functionate 221232
- 6. A generalization of Fatou's theorem 224235
- 7. Remark on functions of the extended class 228239
- 8. Functions orthogonal with respect to the distinguished boundary surface 229240

- BIBLIOGRAPHY 231242
- INDEX 253264