MATHEMATICAL Surveys and Monographs

Volume 5

The Kernel Function and Conformal Mapping Stefan Bergman

http://dx.doi.org/10.1090/surv/005

2000 Mathematics Subject Classification. Primary 30-XX.

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12 11 10 9 8 7 06 05

TO THE MEMORY OF MY SISTERS

Franciszka Anka Bergman

and

Marta Haltrecht

PREFACE TO THE FIRST EDITION

The purpose of this survey is to present a number of methods and principles which are of wide applicability in such branches of analysis as function theory. partial differential equations, differential geometry, etc. The underlying idea is to consider linear classes of functions in which a norm can be introduced and the notion of orthogonality defined. While there is a large degree of arbitrari­ness in the choice of the norm, there exist in each individual case certain natural norms which are distinguished by geometrical or physical properties.

With the help of a complete system of orthonormal functions {<P*(P)} be­longing to a particular class, the kernel function ^L*!°=I<P„(P)*PAQ) of this class is defined. While the formal techniques of operating with complex orthogonal functions are very similar to those well known in the theory of Fourier series, the kernel function is a new concept with fundamental properties which have no counterpart in the classical theory of real orthogonal functions.

The method of complex orthogonal functions can equally well be applied to various fields of analysis such as the theory of functions of several complex variables, the theory of functions satisfying partial differential equations of el­liptic type, and differential geometry. Our special emphasis, however, will be on applications to the theory of conformal mapping. It will be shown that, in the case of multiply-connected domains, the kernel function is intimately related to the classical domain functions, such as the Green's and the Neumann's func­tions, the harmonic measures, and the mapping functions onto canonical do­mains. It thus becomes possible to solve both the boundary value problems of potential theory and the classical conformal mapping problem, once the kernel function of a domain is known.

The fact that the kernel function can be expressed in terms of a complete orthonormal system makes it possible to solve numerically these boundary value and mapping problems for arbitrarily given domains. This is of importance in various fields of physics; in particular, in fluid mechanics elasticity, and elec­tricity. The actual computation of a complete orthonormal system is a rather time-consuming procedure; but with the aid of modern computational technique, these processes come well within the range of practical application.1

Two short chapters, one on partial differential equations of elliptic type and one on functions of two complex variables, have been inserted in the book in order to give the reader a general idea as to how the orthogonal functions method applies to other fields. The reader interested in more detailed information on these subjects is referred to the papers listed in the bibliography.

I wish to express my gratitude to Professors P. R. Garabedian and Zeev Nehari for their cooperation in the preparation of this book, and their numerous valua-

The first step in developing these procedures consists in working out a convenient set of detailed formulas, which later have to be translated into the language of computing machines. See e.g. Bergman [29],

v

vi PREFACE TO THE FIRST EDITION

ble suggestions which resulted in the simplification and more concise presenta­tion of many proofs. Further, I acknowledge my indebtedness to Professor M. Schiffer with whom I discussed large parts of the book and whose contribu­tions to the theory form a substantial part of the material presented. Finally, 1 thank Professor H. Behnke, Dr. L. Geller, Professors M. Heins, H. Royden, L. Sario, and G. Springer for their pertinent criticism and helpful advice while reading the final manuscript.

PREFACE TO THE SECOND EDITION

Various new results about the kernel function, in particular its applications to the geometry of bounded domains, the study of pseudo-conformal trans­formations, l and its abstract approach, have been obtained since the first edition appeared. To make a survey about all these investigations would mean to write a new book, a work which lies beyond the present task and which has to be left to the future. Consequently, various important new results will not be discussed in the present edition. However, the author attempted to compile a complete bibliography, and the reader interested in these (here not discussed) new results should consult the original papers.

The reader will find nonetheless information about some applications outside conformal mapping. In Chapter XI of the first edition some results in the theory of two complex variables are discussed. Since these considerations do not represent a total picture of today's knowledge of these problems, the old Chapter XI has been replaced by new Chapters XI and XII, in which essentially the same questions are treated in the light of present achievements. Again, completeness of the presentation of these problems cannot be claimed.

The modern abstract approach in the theory of Hilbert spaces with the kernel function is not included, since it requires knowledge of functional analysis in greater measure than is generally assumed by the considerations of the present book. Similarly, various results relating to the application of the kernel function to the study of complex manifolds are omitted, since they require acquaintance with differential geometry in a larger degree than expected of the reader.

The contemporary computing machinery opens unlimited possibilities for the numerical applications of mathematical methods in physics and technology. Procedures to be used when applying the theory of the kernel function for the above purposes are indicated in X.5 and various other parts of the book. The author hopes that this new edition will also be of interest to engineers and physicists.

Various chapters of the theory of Hilbert spaces with the kernel functions are presented in other books, in particular lb and the books by Epstein 4a, Fuks5, 5a, 5b, and Meschkowski 15b should be mentioned in this connection. (See p. 231).

I wish to thank my colleagues J. Burbea, P. Caraman, K. T. Hahn, S. Kobayashi, A. Koranyi, C. Loewner, M. Skarczynski and J. A. Wolf for dis­cussions and their valuable advice and suggestions. Thanks are also due to Mrs Charlotte Austin for her outstanding typing and editorial work. Finally the author is indebted to A EC for their support.

1 Pseudo-con formal transform a I ions are one-to-one mappings of domains D by n holomorphic functions of n complex variables with a nonvanishing determinant. Here D denotes a domain of the Z\f *2t * * *,zw-space.

vii

CONTENTS

I. ORTHOGONAL FUNCTIONS 1

1. Introduction, notation, definitions 1 2. The Riesz-Fischer theorem 5 3. The kernel of a system of orthonormal functions g 4. Closed systems 10 5. Doubly orthogonal functions 14 6. A method of analytic continuation lg

II. THE KERNEL FUNCTION AND ASSOCIATED MINIMUM PROBLEMS 21

1. The kernel function 21 2. A general minimum problem . 24

III. THE INVARIANT METRIC AND THE METHOD OF THE MINIMUM INTEGRAL. 31

1. Introduction 31 2. The invariant metric 32 3. Simply-connected domains 34 4. General considerations 35 5. The curvature of the invariant metric 39

IV. KERNEL FUNCTIONS AND HILBERT SPACE . . . . 42

V. REPRESENTATION OF THE CLASSICAL DOMAIN FUNCTIONS 46

1. The Dirichlet integral and the classical functions 46 2. Orthogonal harmonic functions 49 3. Representation of Green's and Neumann's functions 52 4. Further identities involving the kernel function 59 5. Some further domain functions . 62 6. The ^-transforms 67 7. The eigenfunctions of the ^-kernel 71 8. Discussion of the eigenfunctions 75 9. The space A5 and its kernel functions 80 10. Applications to the theory of univalent functions . 86

VI. CANONICAL CONFORMAL TRANSFORMATIONS 89

1. Introduction 89 2. Representative domains. 89 3. An extremal property of the domains £4 95 4. Representation of canonical mapping functions in terms of the kernel function 99 5. Representative domains R(B, i) 105

be

* CONTENTS

VII. ORTHOGONALIZATION OVER THE BOUNDARY 108

1. Definitions and elementary properties 108 2. An auxiliary extremal problem . . ^ 110 3. Identification of the function K(z, £) 114 4. Connections with the theory of bounded functions 117 5. Weighted kernels 120

VIII. VARIATIONAL METHODS , 122

1. The Hadamard variational method 122 2. Properties of monotonicity of domain functions 127 3. The Schiffer variational method 130

IX. EXISTENCE PROOFS 1 3 8

X. PARTIAL DIFFERENTIAL EQUATIONS. 144

1. Introduction - 144 2. Orthogonal expansions for the Green's and Neumann's functions 151 3. Differential equations with non-definite coefficient 155 4. The equation of elasticity 160 5. Numerical evalution of solutions of boundary value problems represented by

the kernel function 164

XL FUNCTIONS OF TWO COMPLEX VARIABLES AND PSEUDO-CONFORMAL MAPPINGS 166

1. Examples of domains and their visualization 166 2. Geometry of the space of two complex variables . > 170 3. Orthogonal functions of two complex variables 176 4. A metric invariant with respect to PCT's * 182 5. Representative domains 187 6. Interior distinguished sets 194 7. The method of minimum integrals 198 8. The behavior of the kernel function KB (Z, Z) in the neighborhood of the

boundary 203 9. Further bounds for the length of the line element of the invariant metric . . . 206

XII. GENERALIZATION OF POTENTIAL-THEORETICAL AND CERTAIN SUB­CLASSES OF FUNCTIONS 211

1. Certain subclasses of functions /(zi ,Z2) in domain with a distinguished boundary surface 211

2. Functions of the extended class 214 3. A generalized Poisson-Jensen formula for analytic polyhedra 216 4. The generalized Nevanlinna formula 219 5. Inequalities for certain functionate 221 6. A generalization of Fatou's theorem 224 7. Remark on functions of the extended class 228 8. Functions orthogonal with respect to the distinguished boundary surface • • • 229

BIBLIOGRAPHY 231

INDEX 253

BIBLIOGRAPHY

BOOKS

1. L. Bieberbach, Lehrbuch der Funktiontheorie, Vols. 1 and 2, Berlin, New York, 1927 and 1945.

l a . L. Ahlfors and L. Sario, Riemann surfaces, Princeton, N . J., 1960. lb. S. Bergman, Integral operators in the theory of linear partial differential equations,

vol. 23, Berlin, 1961 and 1969. lc. S. Bergman and M. Schiffer, Kernel functions and elliptic differential equations in

mathematical physics, New York, 1953. 2. S. Bochner and W. T. Martin, Several complex variables, Princeton, 1948. 2a. H. Bremermann, Distributions, complex variables, and Fourier transforms, Reading,

1965. 2b. P. Caraman, n-dimensional quasiconformal homeomorphisms, Jasi, 1967. (Rumanian) 3. C. Caratheodory, Conformal representation, Cambridge, 1932. 4. R. Courant and D. Hilbert, Methoden der mathematishchen Physik. Vol. I , 2nd rev. ed.,

Berlin, 1931; English transl., Interscience, New York, 1953. 4a. B. Epstein, Orthogonal families of analytic functions, The Macmillan Co., New York,

1965. 5. B. Fuks, Theory of analytic functions of several complex variables, OGIZ, Moscow,

1948. (Russian) 5a , Introduction to the theory of analytic functions of several complex variables,

Fizmatgiz, Moscow, 1962; English transl., Transl. Math. Monographs, vol. 8, Amer. Math. Soc , Providence, R. I., 1963.

5b. , Special chapters in the theory of functions of several complex variables, Fizmatgiz, Moscow, 1963; English transl., Transl. Math. Monographs, vol. 14, Amer. Math. Soc , Providence, R. I., 1965.

5bb. D. Gaier, Konstruktive Methoden der konformen Abbildung, Heidelberg, 1964. 5c. I. M. Gel'fand, D. A. Raikov and G. E. Silov, Commutative normed rings, Moscow,

1960; cf. Amer. Math. Soc. Transl. 5 (1957), 115-220. 5d. R. Gilbert, Function theoretical methods in partial differential equations, Academic

Press, New York, 1969. 5e. S. I. Goldberg, Curvature and homology, New York, 1962. 5f. G. M. Golusin, Geometrical theory of functions of a complex variable, 2nd ed., "Nauka,"

Moscow, 1966. (Russian) 6. E. Goursat, Cours d'analyse mathCmatique, Paris, 1927. 6a. S. Helgason, Differential geometry and symmetric spaces, New York, 1962. 7. D. Hilbert, Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen,

Leipzig-Berlin, 1912. 7a. L. K. Hua, Harmonic analysis of functions of several complex variables in classical

domains, IL, Moscow, 1959; English transl., Transl. Math. Monographs, vol. 6, Amer. Math. Soc., Providence, R. I. 1963.

8. A. Hurwitz and R. Courant, Vorlesungen uber allgemeine Funktionentheorie und elliptische Funktionen, 2 vols., rev. and enlarged ed., Berlin, 1925.

9. G. Julia, Lecpns sur la representation des domaines simplement connexes, Paris, 1941. 10. , Lecpns sur la representation des aires multiplement connexes, Paris, 1934. 11. St. Karczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Warsaw-Lwow, 1935.

231

232 BIBLIOGRAPHY

12. 0 . D. Kellogg, Foundations of potential theory, New York, 1929. 12a. S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 2, Inter-

science Tracts, New York 1963. 13. G. Kowalewski, Einfiihrung in die Determinantentheorie einschliesslich der unend-

lichen und der Fredholmschen Determinanten, Leipzig, 1909. 14. , Integralgleichungen, Berlin and Leipzig, 1930. 14a. E. Kreyszig, Differential geometry, University of Toronto Press, Toronto, 1959. 14b. , Introduction to differential geometry and Riemannian geometry, University

of Toronto Press, Toronto, 1969. 15. P . Levy, Lecons d 'analyse fonctionelle, Paris, 1932. 15a. A. Lichnerowicz, Thiorie globale des connections et de groupes d'holonomie, Rome, 1955. 15b. H. Meschkowski, Hilbertsche Raume mit Kernfunktion, Berlin, 1962. 16. P . Montel, Lecons sur les families normales de fonctions analytiques et leurs applica­

tions, Paris, 1927. 17. , Lecons sur les fonctions univalentes et multivalentes, Paris, 1933. 18. I. D. Murnaghan, Introduction to applied mathematics, New York, 1948. 19. J . v. Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin, 1932. 18a. Z. Nehari, Conformal mapping, McGraw-Hill, 1952. 19a. R. Nevanlinna, La The'orime de Picard-Borel et la thtorie des fonctions mkromorphes,

Paris, 1929. 20. , Eindeutige analytische Funktionen, Berlin, 1936. 20a , Uniformisierung, Berlin, 1953. 21. W. F. Osgood, Lehrbuch der Funktionentheorie, vols. 1 and 2, Leipzig and Berlin,

1929,1938. 21a. J . J. Piateckii-Sapiro, Geometry of classical domains and the theory of automorphic

functions, Moscow, 1961. (Russian) 22. F. Riesz, Les systemes d'equations lintaires a une infinite d'inconnus, Paris, 1913. 22a. B. Rodin and L. Sario, Principal functions, Princeton, N. J., 1968.

22aa.L. Sario and K. Oikawa, Capacity functions, Grundlehren der Math. Wiss., vol. 149, New York, 1969.

22b. D. Spencer and M. Schiffer, Functionals of finite Riemann surfaces, Princeton, N.J. 1954. 23. M. H. Stone, Linear transformations in Hilbert space and their applications to analysis,

Amer. Math. Soc. Colloq. Publ., vol. 15, Amer. Math. Soc , Providence, R. I. 1932. 24. G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer.

Math. Soc , Providence, R. I., 1939. 25. E. C. Titchmarsh, The theory of functions, Oxford, 1932. 25a. V. S. Vladimirov, Methods of the theory of functions of many complex variables,

The M.I.T. Press, Cambridge, Mass., 1966. 26. V. Volterra, Theory of functionals, London, 1931. 27. J . L. Walsh, Interpolation and approximation by rational functions in the complex

domain, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc , Providence, R. I., 1935. 28. A. Weil, Introduction a I'itude de variites kahleriennes, Paris, 1958. 29. K. Yoshida, Functional analysis, Grundlehren d. Math. Wissenschaften, vol. 123,

Berlin-Gottingen-Heidelberg, 1965.

BIBLIOGRAPHY 233

ARTICLES

AHLFORS, L. V. 1. Bounded analytic functions, Duke Math. J., 1947, pp. 1-11. 2. Open Riemann surfaces and extremal problems on compact subregions, Comment.

Math. Helv., vol. 24, 1950, pp. 100-134. 3. Remarks on the Neumann-Poincare integral equation, Pacific J . Math. , vol. 2, 1952,

pp. 271-280. 4. Conformality with respect to Riemann metrics, Ann. Acad. Fenn., Series A, vol. 206, 1955.

ALENICYN, U. E. 1. Conformal mappings of multiply-connected canonical surfaces, Izv. Akad. Nauk Ser.

Mat. SSSR, vol. 28, 1964, pp. 607-644. (Russian) 2. On univalent functions without common values in multiconnected domains, Trudy Mat .

Inst. Steklov, vol. 94, 1968, pp. 1-18. (Russian) Amer. Math. Soc., 1969. ARAVIJSKAJA, E.

Vber ein Verfahren zur effektiven Herstellung von vollstandigen Orthogonalfunktionen-systemen zweier komplexer Veranderlicher, Rec. Math., nouv. ser., vol. 2 (44), 1937, pp. 665-672.

ARENS, R. AND SINGER, J. M. Function values as boundary integrals, Proc. Amer. Math. Soc., vol. 5,1954, pp. 735-745.

ARONSZAJN, N. 1. Sur les invariants des transformations dans le domaine de n variables complexes, C. R.

Acad.Sci., Paris, 1933, pp. 1579-1581. 2. Sur les invariants des transformations dans le domaine de n variables complexes, C. R.

Acad. Sci., Paris, 1934, pp. 143-146. 3. La thiorie des noyaux reproduisants et ses applications. I, Proc. Cambridge Phillos.

Soc., vol. 39, 1943, pp. 133-153. 4. Reproducing and pseudo-reproducing kernels and their application to the partial differ­

ential equations of physics, Technical Report 5, Preliminary Note, Cambridge, Mass., 1948, Navy Contract N5ori 76-16-NR 043-046.

5. Theory of reproducing kernels, Trans. Amer. Math. Soc., vol. 68, 1950, pp. 337-404. 6. Green's functions and reproducing kernels, Proc. Sympos. Spectral Theory and Differ­

ential Problems, Oklahoma Agricultural and Mech. College, Stillwater, Okla., 1951, pp. 355-411.

ARONSZAJN, N. AND SMITH, K. T. 1. A characterization of positive reproducing kernels. Application to Green's functions,

Amer. J. Math., vol. 79, 1957, pp. 611-622. 2. Functional spaces and functional completion. Studies in eigenvalue problems, Report

10, Lawrence. Kansas. 1954. ARONSZAJN, N., MULLA, F. AND SZEPTYCKI, P.

On spaces of potentials connected with Lp classes, Ann. Inst. Fourier (Grenoble), vol. 13, 1963, pp. 211-306.

ARRHEN, P. An integral representation for the four-point function analogous to the Bergman-Weil integral, Nuclear Phys., vol. 81, 1966, p. 557.

BAILY, W. On the imbedding of V-manifolds in projective space, Amer. J. Math., vol. 79, 1957, pp. 403-430.

BEHNKE, H. AND THULLEN Theorie der Funktionen mehrerer komplexer Veraenderlicher, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, 1934.

BERGMAN, S. 1. Ueber die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach

Orthogonal funktionen, Math. Ann., vol. 96, 1922, pp. 237-271 (Thesis, Berlin, 1921).

234 BIBLIOGRAPHY

2. Ueber die Bestimmung der Verzweigungspunkte eines hyperelliptischen Integrals aus seinen Periodizitatsmoduln mil Anwendungen auf die Theorie des Transforms, Math. Z., vol. 19, 1923, pp. 8-25.

3. Ueber die Bestimmung der elastischen Spannungen und Verschiebungen in einem konvexen Koerper, Math. Ann., vol. 98, 1927, pp. 248-263.

3a. Uber die Berechnung der Temperaturverteilung im Rotor einer Dynamomaschine, Z. Angew. Math. Mech., vol. 8, 1928, pp. 402-413.

4. Zwei Satze Uber Funktionen von zwei komplexen Veranderlichen, Math. Ann., vol. 100, 1928, pp. 399-410.

4a. Zur Theorie der algebraischen Potentialfunktionen des dreidimensionalen Raumes, Math. Ann., vol. 99, 1928, pp. 629-659; vol. 101, 1929, pp. 534-558.

5. Ueber die Existenz von Reprasentantenbereichen, Math. Ann., vol. 102,1929, pp. 430-446. 6. Ueber Hermitesche unendliche Formen, die zu einem Bereich gehbren, nebst Anwen­

dungen auf Fragen der Abbildung durch Funktionen zweier komplexen Veranderlichen, Ber. Berliner Math. Ges., vol. 26, 1927, pp. 178-184, and Math. Z., vol. 29, 1929, pp. 640-677.

7. Ueber die schlichten Bereiche in der Theorie der Funktionen von zwei komplexen Ver­anderlichen, J. Reine Angew. Math., vol. 162, 1930, pp. 262-270.

8. Ueber die ausgezeichneten Randflachen in der Theorie der Funktionen von zwei kom­plexen Veranderlichen, Math. Ann., vol. 104, 1931, pp. 611-636.

9. Eine Bemerkung uber schlicht Minimalabbildungen, Ber. Berliner Math. Ges., vol. 30, 1932, pp. 11-13.

10. Ueber den Wertvorrat einer Funktion von zwei komplexen Veranderlichen, Math. Z., vol. 36, 1932, pp. 171-183.

11. Uber eine in der Theorie der Funktionen von zwei komplexen Veranderlichen auftretende unitare Geometrie, Akad. Wetensch. Amsterdam, Proc, vol. 36, 1933, pp. 307-313.

12. Uber die Veranschaulichung der Kreiskorper und Bereiche mit ausgezeichneter Rand-flache, Jber. Deutsch. Math. Ver., vol. 42, 1932, pp. 238-252.

13. Ueber die Kernfunktion eines Bereiches und ihr Verhalten am Rande, J. Reine Angew. Math., vol. 169, 1933, pp. 1-42; vol. 172, 1934, pp. 89-123.1

14. Zwei S&tze aux dem Ideenkreis des Schwarzschen Lemmas Uber die Funktionen von zwei komplexen Veranderlichen, Math. Ann., vol. 109, 1934, pp. 324-348.

14a. Uber eine in gewissen Bereichen mit Maximumflache gUltige Darstellung der Funktionen zweier komplexer Variablen, Math. Z., vol. 39, 1934, pp. 76-94, and pp. 605-608.

14b. Sur quelques proprieUs des transformations par un couple des fonctions de deux variables complexes, Atti. Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur., Ser. 6, vol. 19, 1934, 474-478.

15. Sur les fonctions entieres et miromorphes de deux variables complexes. I, Compositio Math., vol. 3, 1936, pp. 136-173.

15a. Uber eine Integraldarstellung der Funktionen zweier komplexer Veranderlichen, Rec. Math., vol. 1 (43), 1936, pp. 851-862.

16. Zur Theorie von pseudokonformen Abbildungen, Rec. Math., nouv. ser., vol. 1 (43), 1936, pp. 79-96.

17. Zur Theorie der Funktionen, die eine lineare partielle Differentialgleichung befriedigen, Rec. Math., nouv. ser., vol. 2 (44), 1937, pp. 1169-1198.

18. Ueber die Kernfunktion gewisser Reinhardtscher Kreiskorper, Rev. Math. Union Interbalkan., vol. 2, 1939, pp. 41-43.

19. Ueber meromorphe Funktionen von zwei komplexen Veranderlichen, Compositio Math., vol. 6, 1939, pp. 305-335.

20. Theory of pseudo-conformal transformations and its connection with differential geometry, Notes on lectures delivered at the Mass. Institute of Technology, 1939-40.

21. Partial differential equations, Advanced Topics, Brown University, Providence, 1941.

The second part of this paper was at first submitted for publication (about 9 months earlier) to another journal but was rejected by the editor.

BIBLIOGRAPHY 235

22. The method of the minimum integral and analytic continuation of functions of complex variables, Proc. Nat. Acad. Sci. U.S.A., vol. 27, 1941, pp. 328-332.

23. Sur les fonctions orthogonales de plusieur variables complexes avec les applications d la thkorie des fonctions analytiques, Interscience, 1941, and Memor. Sci. Math., vol. 106, 1947.

24. A remark on the paper "Sur les fonctions analytiques de deux variables complexes", J. Math., and Phys., vol. 21, 1942, pp. 141-143.

25. The behaviour of the kernel function at boundary points of the second order, Amer. J. Math., vol. 65, 1943, pp. 679-700.

26. A remark on the mapping of multiply-connected domains, Amer. J. Math., vol. 68, 1946, pp. 20-28.

27. Models in the theory of several complex variables, Amer. Math. Monthly, vol. 53, 1946, pp. 495-501.

28. Functions satisfying certain partial differential equations of elliptic type and their representation, Duke Math. J., vol. 14, 1947, pp. 349-366.

29. Punch-card machine methods applied to the solution of the torsion problem, Quart. Appl. Math., vol. 5, 1943, pp. 69-81.

30. Sur la fonction-noyau d'un domaine et ses applications dans la theorie des transforma­tions pseudo-conformes, Memor. Sci. Math., vol. 108, 1948.

31. Functions of extended class in the theory of functions of several complex variables, Trans. Amer. Math. Soc. vol. 63, 1948, pp. 523-547.

31a. On visualization of domains in the theory of functions of two complex variables, Proc. lnternat. Congr. of Math., vol. 1, 1950, pp. 363-373.

32. A theorem of Green's type for functions of two complex variables, Proc. Amer. Math. Soc , vol. 4, 1953, pp. 102-109.

33. New methods for solving boundary value problems, Z. Angew. Math. Mech., vol. 36, 1956, pp. 182-191.

34. Kernel function and extended classes in the theory of functions of complex variables, Colloque sur les fonction, de plusieurs variables, Bruxelles, 1954, pp. 135-157.

35. Bounds for analytic functions in domains with a distinguished boundary surface, Math. Z., vol. 63, 1955, pp. 173-194.

36. A class of pseudo-con formal and quasi-pseudo -con formal mappings, Math. Ann., vol. 136, 1958, pp. 134-138.

37. A class of quasi-pseudo-con formal transformations in the theory of functions of two complex variables, J. Math. Mech., vol. 7, 1958, pp. 937-956.

38. Les ensembles distinguis dans la theorie des fonctions de deux variables complexes, Rend. Mat. e Appl., vol. 21, 1962, pp. 240-244.

39. Value distribution of meromorphic functions of two complex variables, Studies in Math. Anal, and Related Topics, 1962, pp. 25-37.

40. Distinguished boundary sets in the theory of functions of two complex variables, general topology and its relations to modern analysis and algebra, Proc. Symp. Prague, 1961, pp. 79-86.

40a. Some bounds for coefficients of schlicht pseudo-con formal mappings, Ann. Mat. Pura Appl., vol. 57, 1962, pp. 295-309.

41. The distinguished boundary sets and value distribution of functions of two complex variables, Ann. Acad. Sci. Fenn. Ser. A, I, vol. 12, 1963, pp. 1-28.

42. Some distortion theorems of quasi-analytic mappings in the space of two complex variables, Amer. J. Math., vol. 83, 1963, pp. 405-418.

43. Value distribution of meromorphic functions of two complex variables, Bull. Soc. Sci. Lettres JLodz, vol. 8, 1962, pp. 1-15.

44. Distinguished sets of invariants in the theory of pseudo-con formal transformations, J. Analyse Math., vol. 13, 1964, pp. 317-353.

45. Some properties of pseudo-conformal images of circular domains in the theory of two complex variables, Proc. Minnesota Conference on Complex Analysis, Springer, 1965, pp. 30-40.

236 BIBLIOGRAPHY

46. On boundary behavior of some domain functionals in the theory of functions of two complex variables, Bull. Soc. Sci. Lettres kbdi, vol. 16, 1965, pp. 1-15.

47. On bounds for the distortion of an invariant length under homeomorphism of a bicylinder. Contemporary problems of the theory of analytic functions, "Nauka," Moscow, 1966, pp. 32-44.

48. Interior distinguished points in the theory of functions of two complex variables, J. Analyse Math., vol. 18, 1967, pp. 7-21.

49. Some distortion theorems related to an invariant metric in C2, Colloq. Math., vol. 16, 1966, pp. 67-90.

50. On pseudo-conformal images of circular domains, J. Analyse Math., vol. 22, 1969, pp. 269-284.

51. On some properties of pseudo-con formal images of circular domains, J. Analyse Math. (to appear).

BERGMAN, S., AND CHALMERS, B. A procedure for conformal mapping of triply-connected domains, Math. Comp., vol. 21, 1967, pp. 527-542.

BERGMAN, S., AND HERRIOT, J. G. 1. Application of the method of the kernel function for solving boundary value problems,

Numer. Math., vol. 3, 1961, pp. 209-225. 2. Numerical solution of boundary-value problems by the method of integral operators,

Numer. Math., vol. 7, 1965, pp. 42-65. BERGMAN, S. AND MARCINKIEWICZ, J.

Sur les fonctions analytiques de deux variables complexes, Fund. Math., vol. 33, 1939-1946, pp. 75-94; J. Math, and Phys., vol. 21, 1942, pp. 125-141.

BERGMAN, S. AND MARTIN, W. T. A modified moment problem in two variables, Duke Math. J., vol. 6, 1940, pp. 389-407.

BERGMAN, S. AND SCHIFFER, M. 1. A representation of Green's and Neumann's functions in the theory of partial differential

equations of second order, Duke Math. J., 1947, pp. 609-638. 2. On Green's and Neumann's functions in the theory of partial differential equations,

Bull. Amer. Math. Soc, vol. 53. 1947, pp. 1141-1151. 3. Kernel functions in the theory of partial differential equations of elliptic type, Duke

Math. J., vol. 15, 1948, pp. 535-566. 4. Kernel functions and conformal mapping, Compositio Math., vol. 8, 1951, pp. 205-249. 5. The theory of kernel functions in conformal mapping, Transactions of Symposium on

Conformal Mapping, National Bureau of Standards, Appl. Math. Series 18, 1952, pp. 192-206.

6. Some linear operators in the theory of partial differential equations, Proc. Nat. Acad. Sci. U.S.A., vol. 36, 1950, pp. 742-746.

7. Various kernels in the theory of partial differential equations, Proc. Nat. Acad. Sci. U.S.A., vol. 36, 1950, pp. 559-563.

8. Potential-theoretic methods in the theory of functions of two complex variables, Compositio Math., vol. 10, 1952, pp. 213-240.

BERGMAN, S. AND SPENCER, D. C. 1. On distortion in pseudo-conformal mapping, Trans. Amer. Math. Soc., vol. 51, 1942,

pp. 133-163. 2. A property of pseudo-conformal transformations in the neighborhood of boundary points,

Duke Math. J., vol. 9, 1942, pp. 757-762. BERS, L.

1. Sur une representation integrate des fonctions biharmoniques dans les domaines possidant une surface frontiire remarquable, C. R. Acad. Sci. Paris, vol. 208, 1939, pp. 1273-1275.

2. Sur les valeurs limites des fonctions analytiques de deux variables complexes dans les domaines possidant une surface frontUre remarquable, C. R. Acad. Sci. Paris, vol. 208, 1939, pp. 1475-1477.

BIBLIOGRAPHY 237

3. On bounded analytic functions of two complex variables in certain domains with dis­tinguished boundary surfaces, Amer. J . Math., vol. 64, 1942, pp. 514-529.

BlEBERBACH, L. Zur Theorie und Praxis der konformen Abbildung, Rend. Circ. Mat. Palermo, vol. 38, 1914, pp. 98-112.

BlERNACKl, M. Sur quelques majorantes de la thiorie des fonctions univalents, C. R. Acad. Sci. Paris, vol. 201, 1935, pp. 256-258.

BISHOP, E. A minimal boundary for function algebras, Pacific J. Math., vol. 9, 1959, p. 629.

BJORKEN, J. D. Spectral representation of Green's function in perturbation theory, Thesis, Stanford University, 1959.

BLOCK, I. E. Kernel functions and class L , Proc. Amer. Math. Soc , vol. 4, 1953, pp. 110-117.

BOCHNER, S. 1. Ueber orthogonale Systeme analytischer Funktionen, Math. Z., vol. 14, 1922, pp. 180-207

(Thesis, Berlin, 1921). 2. Functions of integrable square in several complex variables, Duke Math. J., vol. 4,

1938, pp. 686-706. 3. Boundary values of analytic functions in several variables and of almost periodic func­

tions, Ann. of Math., vol. 45, 1944, pp. 708-722. BREMERMANN, H. J.

1. Die Holomorphiehullen der Tuben- und Halbtubengebiete, Math . Ann., vol. 127, 1954, pp. 406-423.

2. Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, Lectures on Functions of a Complex Variable, Univ. of Michigan Press, Ann Arbor, Mich., 1955, pp. 349-383.

3. Complex convexity, Trans. Amer. Math. Soc , vol. 82, 1956, pp. 17-51. 4. Holomorphic functional^ and complex convexity in Banach spaces, Pacific J. Math.,

vol. 7, 1957, pp. 811-831. 5. On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex

domains, Trans. Amer. Math. S o c , vol. 91, 1959, pp. 246-276. BURBEA, J.

1. On the kernel function in domains with Bergman-Silov boundary surface. J. Math. and Mech. (to appear).

2. Sur la fonction-noyau dans des domaines ayant une surface frontiire de Bergman-Silov, Comptes Rendus Acad. Sci., Paris, vol. 269, 1969, pp. 344-346.

3. On the Gaussian curvature related to the S2ego metric (to appear). 4. A procedure for conformal maps of simply connected domains by using the Bergman

function (to appear). CALDERON, A. P. AND ZYCMUND, A.

Note on boundary values of functions of several complex variables, Ann. of Math. Studies, no. 25, Princeton Univ. Press, Princeton, N. J., 1950.

CARAMAN, P. 1. Quelques proprietis des representations quasi-conformes de En. I, Rev. Roumaine

Math. Pures Appl., vol. 10, 1965, pp. 180-181. 2. A new definition of the n-dimensional quasiconformal mappings, Nagoya Math. J.,

vol. 26, 1966, pp. 145-165 3. About the connection between the 2n-dimensional quasiconformal homeomorphisms

(QCfH) and the n-dimensional pseudo-conformal transformations (PCT), Rev. Roumaine Math. Pures Appl., vol. 13, 1968, pp. 1255-1271.

CARATHEODORY, C. Uber die Geometrie der analytischen Abbildungen, Abh. Math. Sem. Univ. Hamburg, vol. 6, 1928, pp. 96-145.

238 BIBLIOGRAPHY

CARLEMAN, T. Ueber die Approximation analytischer Funktionen durch lineare Aggregate von vorgege-benen Potenzreihen, Ark. Mat. Astr. Fys., vol. 17, 9, 1923, 30 pp.

CARTAN, E. Sur les domaines bornes, homogenes de I'espace de n variables complexes, Abh. Math. Sem. Univ. Hamburg, vol. 11, 1936, pp. 116-162.

CARTAN,H. 1. Les fonctions de deux variables complexes et le problime de la representation analytique,

J. Math. Pures Appl., vol. 10, 1931, pp. 1-114. 2. Sur les groupes de transformations analytiques, Actualites Sci. Ind., Exposes Math.

IX, Paris, 1935. CHALMERS, B.

1. Surle comportement, 6 la frontiire, de la fonction-noyau de Bergman, C. R. Acad. Sci., Paris, vol. 266, 1968, pp. 1132-1134.

2. On boundary behavior of the Bergman kernel function and related domain Junctionals, Pacific J. Math., vol. 29, 1969, pp. 243-250.

3. Subspace kernels and minimum problems in Hilbert spaces with kernel function, Pacific J. Math., vol. 31, 1969, pp. 619-628.

4. Linear transformations in Hilbert spaces with kernel function (to appear). CHARZYNSKI, Z.

Bounds for analytic functions of two complex variables, Math. Z., vol. 75, 1961, pp. 29-35. CHERN,S. S.

On holomorphic mappings of hermitian manifolds of the same dimension, vol. 11, Proc. Sympos. Pure Math., Amer. Math. Soc, Providence, R. I., vol. 11, 1968, pp. 157-170.

CHEVALIER, H. Noyau sur une varieU analytique, C. R. Acad. Sci. Paris, vol. 262, 1966, pp. 948-950

CIMA, J., AND COVER. A. Some results on HP(D2), J. Analyse Math., vol. 22, 1969, pp. 333-341.

COHN, H. How perturbations satisfying Hilbert s modular group affect the invariant metric, J. Analyse Math., vol. 18, 1967, pp. 53-59.

COLTON, D . AND GILBERT, R. P . Rapidly convergent approximations to Dirichlet s problem for semilinear elliptic equations (to appear).

COURANT, R . , MANEL, B . AND SHIFFMAN, M . A general theorem on conformal mapping of multiply connected domains, Proc. Nat. Acad. Sci. U.S.A., vol. 26, 1940, pp. 503-507.

DAVIS, P. J. 1. Errors of numerical approximation for analytic functions, J. Rational Mech. Anal.,

vol. 2, 1953, pp. 303-313. 2. An application of doubly orthogonal functions to a problem of approximation in two

regions, Trans. Amer. Math. Soc, vol. 72 (1954), pp. 104-137. DAVIS, P. J. AND POLLAK, H.

On the analytic continuation of mapping functions, Trans. Amer. Math. Soc, vol. 87, 1958, pp. 198-225.

DAVIS, P. J. AND RABINOWITZ, P. 1. Advances in orthonormalizing computation, National Bureau of Standards, Washington,

D.C.,0000. 2. Numerical experiments in potential theory using orthonormal functions, J. Wash. Acad.

Sci., vol. 46, 1956, pp. 12-17. DAVIS, P. J. AND WALSH, J. L.

1. Interpolation and orthonormal systems, J. Analyse Math., vol. 2, 1952, pp. 1-28. 2. On representations and extensions of bounded linear junctionals defined on classes of

analytic functions, Trans. Amer. Math. Soc, vol. 76, 1954, pp. 190-206. DEETER, C. R. AND SPRINGER, G.

Discrete harmonic kernels, J. Math. Mech., vol. 14, 1965, pp. 413-438.

BIBLIOGRAPHY 239

DERRICK, W. A volume-diameter-inequality for n-cubes, J. Analyse Math., vol. 22, 1969, pp. 1-36.

DlEDERlCH, K. Das Randuerhalten der Bergmanschen Kernfunktion und Metrik in strong pseudo-konvexen Gebieten, Math. Ann. (to appear).

DINGHAS, A. 1. Ein n -dimensionales Analogon des Schwarz-Pickschen Satzes fur holomorphe A bbildungen

der komplexen Einhcitskugel in eine Kahler-Mannigfaltigkeit, Weierstrass Festschrift, Westdeutscher Verlag, Koln, 1965, pp. 477-494.

2. Vber das Schwarzsche Lemma und verwandte Satze, Israel J. Math., (to appear). EARLE, C. J.

1. A reproducing formula for integrable automorphic forms, Amer. J. Math., vol. 88, 1966, pp. 867-870.

2. Some remarks on Poincane series, Comp. Math., vol. 21, 1969, pp. 167-176. EDEN, R. J.

Threshold behaviour in quantum field theory, Proc. Roy. Soc. Ser. A, vol. 210, 1952, pp. 388-404.

EGOROV, I. P. On a class of kernel functions, Volz. Mat. Sb., vol. 3, 1965, 145-148. (Russian)

EKE, B. Value distribution of regular functions of two complex variables, Ann. Polon. Math. (to appear).

EVANS, G. C. Sur I'inttgrale de Poisson, Comptes Rendus, vol. 177, 1923, pp. 241-242.

EVANS, G. C. AND BRAY, H. E. 1. Sur I 'integrate de Poisson gentraliste, Comptes Rendus, vol. 176, 1923, pp. 1042-1044. 2. La formule de Poisson et le problime de Dirichlet, Comptes Rendus, vol. 176, 1923,

pp. 1368-1371. FARRELL, O.

On approximation to analytic functions by polynomials, Bull. Amer. Math. Soc, vol. 40, 1940, pp. 908-914.

FATOU, P. Siries trigonome'trique et series de Taylor, Acta Math., vol. 30, 1906, pp. 335-400.

FlCHERA, G. Sulla ''kernel function". Boll. Un. Mat. Ital., Ser. Ill, vol. 7, 1952, pp. 4-15.

FRONSDAL, C, MAHANTAPPA, K. AND NORTON, R. 1. Integral representation for a scattering amplitude with complex singularities, Phys.

Rev., vol. 127, 1962, p. 1847. 2. More integral representations for scattering amplitudes with complex singularities,

J. Mathematical Phys., vol. 4, 1963, p. 859. FUKS, B.

1. Limitations pour la variation d'un angle dans le cas d'une transformation pseudo-conforme dans I'espace de deux variables complexes, C. R. Acad. Sci. Paris, vol. 200, 1935, pp. 718-720.

2. Ueber einige Eigenschaften der pseudokonformen Abbildungen, Rec. Math., nouv. ser., vol. 1 (43), 1936, pp. 569-574.

3. Ueber die Gruppe der pseudokonformen Abbildungen eines Bereiches in sich, Bull. Inst. Math, et Mec. Univ. de Tomsk, vol. 1, 1935—1937, pp. 281-285.

3a. Group of motions of a geometry invariant under pseudo-conformal mappings, Dokl. Akad. Nauk SSSR, vol. 16, 1937, pp. 143-146. (Russian)

4. Zur Theorie der schlichten pseudokonformen Abbildungen, C. R. Acad. Sci., URSS, nouv. ser., vol. 20, 1938, pp. 3-4.

4a. Locally isometric mappings, Dokl. Akad. Nauk SSSR, vol. 18, 1938, pp. 3-4. (Russian) 5. Ueber eine Eigenschaft der bei pseudokonformen Abbildungen invarianten Metrik, Rec.

Math., nouv. ser., vol. 5 (47), 1939, pp. 497-504.

240 BIBLIOGRAPHY

6. Sur les representations pseudo-conformes a"un domaine sur lui-mime avec un point fixe sur la frontiere, C. R. Acad. Sci., URSS, vol. 23, 1939, pp. 497-504.

7. Vber geodatische Mannigfaltigkeiten einer bei pseudokonformen Abbildungen invarianten Riemannschen Geometric Rec. Math., nouv. ser., vol. 2 (44), 1937, pp. 567-594.

7a. Minimum function of a domain, Mat. Sb. (N.S.), vol. 16 (58), 1945, pp. 21-38; vol. 18 (60), 1946, pp. 329-345. (Russian)

8. On the Ricci curvature of Bergmans metric, invariant with respect to biholomorphic transformations, Dokl. Akad. Nauk SSSR, vol. 167, 1966, pp. 996-999. (Russian)

9. Metric invariant with respect to biholomorphic transformation, Abstracts of Lectures, Internat. Congr. of Mathematicians, 10th Section, 1966. (Russian)

GAL, I. S. Conformally invariant metrics and uniform structures, Nederl. Akad. Wetensch. Proc. Ser. A, vol. 63, 1960, pp. 218-244.

GARABEDIAN, P. 1. Schwarz' lemma and the Szego kernel function, Trans. Amer. Math. Soc, vol. 67,

1949, pp. 1-35. 2. Distortion of length in conformal mapping, Duke Math. J., vol. 16, 1949, pp. 439-459. 3. A partial differential equation arising in conformal mapping, Pacific J. Math., vol. 1,

1951, pp. 485-524. 4. A new proof of the Riemann mapping theorem, Proc. of a Symposium on Conformal

Mapping, National Bureau of Standards, Appl. Math. Ser., vol. 18, 1952, pp. 207-214. 5. A new formalism for functions of several complex variables, J. Analyse Math., vol. 1,

1951, pp. 59-80. 6. Orgononal harmonic polynomials, Pacific J. Math., vol. 3, 1953, pp. 585-603.

GARABEDIAN, P. AND SCHIFFER, M. 1. Identities in the theory of conformal mapping, Trans. Amer. Math. Soc, vol. 65,

1949, pp. 187-238. 2. On existence theorems of potential theory and conformal mapping, Annals of Math.,

vol. 52, 1950, pp. 164-187. GEHRING, F. W.

Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc, vol. 103, 1962, pp. 353-393.

GEHRING, F. W. AND VAISALA, J. The coefficients of quasiconformality of domains in space, Acta Math., vol. 114, 1965, p. 21.

GELBART, A. On the growth of properties of a function of two complex variables given by its power series exoansion, Trans. Amer. Math. Soc, vol. 59, 1941, pp. 199-210.

GELFAND, I. M., RAIKOV, D. A. AND SILOV, D. £. Commutative normed rings, Uspehi Mat. Nauk 1. No. 2 (12), 1946, pp. 48-146.

GILBERT, R. P. 1. A new method for solving boundary value problems, Bull. Amer. Math. Soc. (to appear). 2. Singularities of three-dimensional harmonic functions, Carnegie Institute of Tech­

nology, 1958. 3. Singularities of three-dimensional harmonic functions, Pacific J. Math., vol. 10, 1960,

pp. 1243-1255. 4. Singularities of solutions to the wave-equation in three dimensions, J. Reine Angew.

Math., vol. 205, 1960, pp. 75-81. 5. A generalization of Hadamards singularity condition, (to appear).

GINDIKIN, S. G., PJATECKII-SAPIRO, 1.1, AND VINBERG, £. B. 1. On classification and canonical realization of complex, homogeneous bounded domains

Trudy Moskov. Mat. Obsc, vol. 12, 1963. (Russian) 2. Homogeneous Kahler manifolds, Geometry of homogeneous bounded domains, Centro

Internationale Matematico Estivo (C. I. M. E.) Roma, 1968, pp. 1-88.

BIBLIOGRAPHY 241

GLOCK, M. On the Sibv boundary of the vertex function. I and I I , Thesis, Haifa, 1966, (to appear).

GODEMENT, R. Fonctions holomorphes de carri sommable dans le de mi-plan de Siegel, Seminaire de Henri Cartan, lOe annee 1957/58, No. 6, Ecole Normal Superieure, 1958.

GOLDSTEIN, M. K- and L-kernels on an arbitrary Riemann surface, Pacific J. Math. , vol. 19, 1966, pp. 449-459.

GREENSTONE, L. Mapping by analytic functions. Par t I: Conformal mapping of multiply-connected domains, Trans. Amer. Math. Soc. vol. 63, 1948, pp. 125-143.

GROETZSCH, H. 1. Ueber einige Extremalprobleme der konformen Abbildung. I , I I , Ber. Akad. Leipzig,

vol. 80, 1928, pp. 367-376, and 497-502. 2. Ueber die Verzerrung bei schlichter konformer Abbildung mehrfachzusammenhangender

schlichter Bereiche. II, Ber. Akad. Leipzig, vol. 81, 1929, pp. 217-221. GRUNSKY, H.

1. Neue Abschatzungen zur Theorie der konformen Abbildung ein- und mehrfachzusam­menhangender Bereiche, Schr. Math. Sem. Univ., Berlin, vol. 1, 1932, pp. 95-140.

2. Eindeutige beschrankte Funktionen in mehrfachzusammenhangenden Gebietin, Jbr . Deutsch. Math. Ver., vols. 50 and 52, 1940, 1942, pp. 230-235 and 118-132.

3. Koeffizientenbedingungen fur schlicht abbildende meromorphe Funktionen, Math. Z., vol. 45, 1939, pp. 29-61.

HADAMARD, J. 1. M&moire sur le probleme d analyse relatif a I'iquilibre de plaques elastiques encastrees,

Memoirs des savants etrangers, vol. 33, 1908. 2. Lecons sur le calcul des variations, Paris. 1910. 3. Theoreme sur les series entieres, Acta Math., vol. 22, 1898, pp. 55-63.

HAHN, K.T. 1. Minimum problems of Plateau type in the Bergman metric space, Pacific J. Math.,

vol. 14, 1964, pp. 943-955. 2. The non-euclidean Pythagorean theorem with respect to the Bergman metric, Duke Math.

J., vol. 33, 1966, pp. 523-534. 3. Some properties of relative invariants on bounded domains, Duke Math. J., vol. 34, 1967,

pp. 325-332. 4. A remark on integral functions of several complex variables, Pacific J . Math., vol.

26, 1968. pp. 509-513. 5. Trigonometry in a hyperbolic space, Duke Math. J., vol. 35, 1968, pp. 739-745. 6. Note on linear functional in Hp spaces, Canad. J . Math , (to appear).

HAHN, K. T. AND MITCHELL, J. 1. Generalization of Schwarz-Pick lemma to invariant volume in a Kahler manifold,

Trans. Amer. Math. Soc, vol. 128, 1967, pp. 221-231. 2. Generalization of Schwarz-Pick lemma to invariant volume in a Kahler manifold. I I , Bull.

Amer. Math. Soc , vol. 73, 1967, pp. 668-670. 3. Green's functions on the classical Cartan domains, MRC Tech. Summary Report No.

800, Univ. of Wisconsin, Dec. 1967. 4. Generalization of Schwarz-Pick lemma to invariant volume, Canad. J . Math. , vol. 21,

1969, 669-674. 5. Hp spaces on bounded symmetric domains, Trans. Amer. Math. Soc. vol. 146, 1969,

pp.521-531. 6. Convergence in Hp spaces on bounded symmetric domains, Pacific J . Math, (to appear).

HAMMERSTEIN, A. Vber die Approximation von Funktionen zweierkomplexer Veranderlichen durch Polynome, S.-B. Preuss. Akad. Wiss., 1933, pp. 259-266.

HAWLEY, N . S. AND SCH1FFER, M. 1. Connections and conformal mapping, Acta Math., vol. 107, 1962, pp. 175-274.

242 BIBLIOGRAPHY

2. Half-order differentials on Riemann surfaces, Acta Math., vol. 115, 1966, pp. 199-236. 3. Riemann surfaces which are doubles of plane domains, Pacific J. Math., vol. 20,

1967, pp. 217-222. HEFER, H.

Zur Funktionentheorie mehrerer Veranderlichen. Vber eine Zerlegung analytischer Funktionen unddie Weilsche Integraldarstellung, Math. Ann., vol. 122, 1950, pp. 276-278.

HEINS, M. A Lemma on positive harmonic functions, Ann. of Math., vol. 52, 1950, pp. 568-573.

HELLINGER, E. AND TOPLITZ, 0 . Integralgleichungen und GLeichungen mit unendlich vielen Unbekannten, Enz. Math. Wiss., II, 3.

HERGLOTZ, G. Vber Potenzreihen mit positivem Realteil im Einheitskreise, Leipziger Berichte, vol. 63, 1911, pp. 501-511.

HlGUCHI,T. On the Riemannian curvature and the Poisson integral in several complex variables, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A, vol. 7, 1961, pp. 34-45.

HlGUCHI, T . AND HlGUCHl, Y. On the continuation of the kernel form in a certain class of complex manifolds, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A. vol. 9, 1965, pp. 77-86.

HITOTUMATU, S. On quasi-con formal functions of several complex variables, J. Math. Mech., vol. 8, 1959, pp. 77-94.

HOFFMAN, K. Minimal boundaries for analytic polyhedra, Rend. Circ. Mat. Palermo, series II, vol. 9, 1960, pp. 1-14.

HORMANDER, L. Existence theorem for the 6-operator by L2 methods, Acta Math., vol. 113, 1965, pp. 89-152.

HsO, I-CH'AO On the classification of symmetric schlicht domains in several complex variables, Shuxue Jinzhan, vol. 8, 1965, pp. 109-144.

INNIS, G. S. Some reproducing kernels for the unit disk, Pacific J. Math., vol. 14, 1964, pp. 177-186.

ISLAM, J. N. Modified Mandelstam representation for heavy particles, J. Mathematical Phys., vol. 3, 1962, p. 1098.

JURASEV, V. V. 1. On the Ricci curvature of a geometry defined in a domain (fn by a sequence of holomorphic

functions, Moskov. Oblast. Ped. Inst. Ucen. Zap., 1967 (Russian) (to appear). 2. On the Ricci curvature of the geometry defined in a domain D C Cn with the aid of a

system of analytic functions, Izv. Akad. Nauk Armjan SSR Ser. Mat., vol. 3 (No. 1), 1968, pp. 70-74. (Russian)

3. Estimates of the Riemann and Ricci curvatures of Bergman's metric, Trudy Mat. Inst. Steklov, vol. 4, 1968, pp. 30-52. (Russian)

KAHLER, E. Vber eine bemerkenswerte Hermitesche Metrik, Abh. Math. Sem. Univ. Hamburg, vol. 9, 1933, pp. 173-186.

KAKURAI, S. Characterization of canonical domains of Bergman and analytic automorphisms with an invariant point, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, vol. 7, 1963, pp. 275-307.

KALLEN, G. "Properties of vacuum expectation values of field operators", in Relations de dispersion et particules elementaires, Ecole d'ete de physique theorique, Les H ouches, 1960.

BIBLIOGRAPHY 243

KALLEN, G. AND TOLL, J. 1. Integral representation for the vacuum expectation value of three scalar local fields,

Helv. Phys. Acta, vol. 33, 1960, p. 753. 2. Momentum space analyticity properties of the Bergman-Weil integral for the three

point function, J. Mathematical Phys., vol. 6, 1965, p. 1885. KALLEN, G. AND WlGHTMAN, A.

The analytic properties of the vacuum expectation value of a product of three scalar local fields, Mat.-Fys. Skr. Danske Vid. Selsk, vol. 1, 1958.

KAMBARTEL, F. Orthonormale Systeme und Randintegralformeln in der Funktionentheorie mehrerer Veranderlichen, Schr. Math. Inst. Univ. Munster, vol. 18, 1960.

KANEMARU, T. On the A-representative domain, Sci. Rep. Tokyo Kyioku Daigaku Sect. A, vol. 8, 1965, pp. 259-270.

KATO, S. Canonical domains in several complex variables, Pac. J . Math. , vol. 21,1967, pp. 279-291.

KlERNAN,P. 1. Hyperbolic submanifolds of complex projective space, Proc. Amer. Math. Soc., vol. 22,

1969, pp. 603-606. 2. On the relations between taut, tight and hyperbolic manifolds, Bull. Amer. Math. Soc.,

vol. 76, 1970, pp. 49-51. 3. Quasiconformal mappings and Schwarz's lemma, Trans. Amer. Math. Soc., 1970,

(to appear). 4. Some remarks on hyperbolic manifolds, Proc. Amer. Math. Soc. (to appear).

KOBAYASHI, S. 1. Geometry of bounded domains, Trans. Amer. Math. Soc., vol. 92, 1959, pp. 267-290. 2. On automorphism groups of homogeneous complex manifolds, Proc. Amer. Math. S o c ,

vol. 12, 1961, pp. 359-361. 3. On complete Bergman metrics, Proc. Amer. Math. Soc , vol. 13, 1962, pp. 511-513. 4. Volume elements, holomorphic mappings and Schwarz's lemma, Proc. sympos. Pure

Math., vol. 11, Amer. Math. Soc , Providence, R. I., 1968. (to appear) 5. Intrinsic metrics on complex manifolds, Bull. Amer. Math. Soc , vol. 73,1967, pp. 347-349. 6. Distance, holomorphic mappings and the Schwarz's lemma, J . Math. Soc , Japan,

(to appear). KODAIRA,K.

On Kahler varieties of restricted type {an intrinsic characterization of algebraic varieties), Ann. of Math., Vol. 60, 1954, pp. 28-48.

KOEBE, P. Abhandlungen zur Theorie der konformen Abbildung, IV, Acta Math., vol. 41, 1920, pp. 235-301.

KORANYI, A. 1. The Bergman kernel function for tubes over convex cones, Pacific J . Math. , vol. 12 (1962),

pp. 1355-1359. 2. The Poisson integral for generalized half-planes and bounded symmetric domains,

Ann. of Math., vol. 82, 1965, pp. 332-350. 3. A Schwarz lemma for bounded symmetric domains, Proc. Amer. Math. Soc , vol. 17,

1966, pp. 210-213. 4. Analytic invariants of bounded symmetric domains, Proc. Amer. Math. Soc , vol. 19,

1968, pp. 279-284. 5. Generalizations of Fatou's theorem to symmetric spaces, to appear. 6. Holomorphic and harmonic functions on bounded symmetric-domains, Geometry of

homogeneous bounded domains, Centro Internationale Matematico Estivo (C. 1. M. E.) Roma, 1968, pp. 125-196.

7. Harmonic functions on Hermitian hyperbolic spaces, Trans. Amer. Math. S o c , vol. 135, 1969, pp. 507-516.

244 BIBLIOGRAPHY

KORANYI, A. AND WOLF, J. A. 1. Realization of hermitian symmetric spaces as generalized half planes, Ann. of Math.,

vol. 81, 1965, pp. 265-288. 2. Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math.,

vol. 87, 1965, pp. 899-939. KOSZUL, J. L.

1. Sur la forme hermitienne canonique des espaces homogines complex, Canad. J. Math., vol. 7, 1955, pp. 562-576.

2. Exposes sur les espaces homogines symHriques, University of Sao Paulo. 1959. KREYSZIC, £ .

1. On a class of partial differential equations, J. Rational Mech. Anal., vol. 4, 1955, pp. 907-923.

2. On coefficient problems of solutions of partial differential equations of the fourth order, J. Math. Mech., vol. 6, 1957, pp. 811-822.

3. On regular and singular harmonic functions of three variables, Arch. Rational Mech. Anal., vol. 4, 1960, pp. 352-370.

KUFAREFF, P . Ueberdas zweifach zusammenhangende Minimalgebiet, Bull. Inst. Math, et Mec. Univ. de Tomsk, vol. 1, 1935-1937, pp. 228-236.

KWACK, U. H.

Generalization of the big Picard theorem, Ann. of Math. Vol. 90, 1969, pp. 9-22. LANDAU, L. D.

1. On analytic properties of vertex parts in quantum field theory, Nuclear Phys., vol. 13, 1959, pp. 181-192.

2. On the analytic properties of vertex parts in quantum field theory, Soviet Physics JETP, vol. 37, 1960, pp. 45-50.

LEBED, B. J. Some bounds for the Riemann curvature of an invariant metric, Trudy Moskov. Inst. Electron. Masinostroenija, vol. 4, 1967. (Russian) (to appear)

LEBED, B. J. AND JURASEV, V. V. 1. On the curvature of an invariant metric, Trudy Moskov. Inst. Electron. Masinostroenija,

vol. 4, 1967. (Russian) (to appear) 2. On an estimate of J. Stark, Trudy Mat. Inst. Steklov, vol. 4, 1968, pp. 53-55.

LEHTO, O. Anwendung orthogonaler Systeme auf gewisse funktionentheoretische Extremal- und AbbildungS'probleme, Helsinki, 1949.

LELONG.P. 1. Les fonctions plurisousharmoniques, Ann. Sci. Ecole Norm. Sup. (3), vol. 62, 1945,

pp. 301-338. 2. Domaines convexes par rapport aux fonctions plurisousharmoniques, J. Analyse Math.,

vol. 2, 1952, pp. 178-208. 3. Fonctions plurisousharmoniques; mesures de Radon associees. Applications aux fonctions

analytiques. Colloque sur les fonctions de plusieurs variables, tenu a Bruxelles, 1953, pp. 21-40, Masson, Paris, 1953.

4. Ensembles singuliers impropres des fonctions plurisousharmoniques, J. Math. Pures Appl. (9), vol. 36, 1957, pp. 263-303.

LERAY, J. Calcul differential et inUgral sur une vartitt analytique complexe {Probleme de Cauchy, III) , Bull. Soc. Math. France, vol. 87, 1959, pp. 81-180.

LlCHNEROWICZ, A. 1. Varietts analytiques complexes et tenseur de Bergmann, C. R. Acad. Sci. Paris, vol.

259, 1964, pp. 2737-2740; vol. 261, 1965, pp. 2143-2145. 2. Varittis complexes et tenseur de Bergmann, Ann. Inst. Fourier (Grenoble), vol. 15,

1965, pp. 345-408.

BIBLIOGRAPHY 245

3. VariiUs analytiques complexes et tenseur de Bergmann, Simposio Internationale di Geometrica algebraica Roma 1965, Edizioni Cremonese Roma 1967, pp. 113-128.

LlCHTENSTEIN, L. Neuere Entwicklung der Theorie der partiellen Differentialgleichungen zweiter Ordnung von elliptischem Typus, Enz. Math. Wiss., vol. II, 3, 1.

Lo,C. Y. Bergman's kernel function on a class of elliptic partial differential equations in four variables, Ann. Mat. Pura Appl., vol. 79, 1968, pp. 93-105.

LOEWNER, C. On the conformal capacity in space, J. Math. Mech., vol. 8, 1959, pp. 411-414.

LOKKI, O. Ueber Existenzbeweise einiger mit Extremaleigenschaft verschenen analytischen Funk-tionen, Ann. Acad. Sci. Fenn. Series A, Number 76, 1950. 15 pp.

LOOK, K. H. ( L U QI-KENG) 1. Schwarz lemma in the theory of functions of several complex variables, Sci. Rep., New

Series, vol. 1, 1957, pp. 5-9. 2. Schwarz lemma and analytic invariants, Sci. Sinica, vol. 7, 1953, pp. 453-504. 3. On Kaehler manifolds with constant curvature, Acta Math. Sinica, vol. 16, 1966,

pp. 269-281. 4. An analytic invariant and its characteristic properties, Acta Math. Sinica, vol. 8, 1958,

pp. 243-252. LOWDENSLAGER, D. B.

1. Potential theory and a generalized Jensen-Nevanlinna formula for functions of several complex variables, J. Math. Mech., vol. 7, 1958, pp. 207-218.

2. Potential theory in bounded symmetrical homogeneous complex domains, Ann. of Math., vol. 67, 1958, pp. 467-484.

Lu,Y.-C. Holomorphic mappings of complex manifolds, J. Differential Geometry, vol. 2, 1968, pp. 299-312.

MARTIN, W. T. Minimum problem in the theory of analytic functions of several variables, Trans. Amer. Math. Soc , vol. 48, 1940, pp. 350-358.

MASCHLER, M. 1. Minimal domains and their Bergman kernel function, Pacific J . Math. , vol. 6, 1956,

pp. 501-516. 2. Classes of minimal and representative domains and their kernel functions, Pacific J .

Math., vol. 9, 1959, pp. 763-782. 3. Analytic functions of the classes L2 and I2 and their kernel functions, Rend. Circ. Mat .

Palermo, vol. 8, 1959, pp. 1-15. 4. Derivatives of the harmonic measures in multiply-connected domains, Pacific J . Math. ,

vol. 12, 1962, pp. 637-647. MATSUURA, S.

1. On the theory of pseudo-conformal mappings, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A. vol. 7, 1963, pp. 231-253.

2. On the normal domains and the geodesies in the bounded symmetric spaces and the projective space, Sci. Rep. of the Gunma Univ., vol. 15, 1966, pp. 1-21.

MEHRING, J. Kernfunktion und Regularitatsgebiete im Raum vom zwei komplexen Veranderlichen, Dissertation, Munster, 1953.

MESCHKOWSKI, H. 1. Verzerrungssatze fur mehrfach zusammenhangende Bereiche, Compositio Math.,

vol. 11, 1953, pp. 44-59. 2. Beitrage zur Theorie der Orthonormalsysteme, Math . Ann., vol. 127, 1943, pp. 107-129. 3. Verallgemeinerung der Poissonschen Jntegralformel auf mehrfach zusammenhangende

Bereiche, Ann. Acad. Sci. Fenn. Ser. A, I, 1954, p. 166.

246 BIBLIOGRAPHY

4. Die Koeffizienten des Bergmanschen Orthonormalsystems, Math. Ann., vol. 128, 1954, pp. 200-203.

5. Vber Hilbertsche Raume mit Kernfunktion, Arch. Math., vol. VI, 1955, pp. 151-156,481. 6. Darstellung analytischer Funktionen durch den Randwinkel des Biidbereichs, Math.

Z., vol. 66, 1955, pp. 189-202. 7. Interpolation durch Funktionen eines Orthonormalsystems, Arch. Math., vol. VII,

1957, pp. 175-179. 8. Zur Theorie der Interpolationsreihen, J. Reine Angew. Math., vol. 202, 1959, pp. 9-15. 9. Vber die konforme Abbildung gewisser Bereiche von unendlich hohem Zusammenhang

auf Vollkreise, Math. Ann., vol. 123, 1951, pp. 392-405; vol. 124, 1952, pp. 178-181. 10. Beziehungen zwischen den Normalabbildungsfunktionen in der Theorie der konformen

Abbildung, Math. Z., vol. 55, 1951, pp. 114-124. 11. Einige Extremalprobleme aus der Theorie der konformen Abbildung, Ann. Acad. Sci.

Fenn.Ser. Al, vol. 117, 1952, pp. 3-12. MICHIWAKI, Y.

1. On the theory of analytic functions of several complex variables, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A, vol. 6, 1959, pp. 307-324.

2. Remarks on the theory of analytic functions of several complex variables, Res. Rep. Nagaoka Tech. College, vol. 1, 1962, pp. 3-9.

3. The kernel functions and associated minimum problems, Riemannian curvature in an analytic direction, Res. Rep. Nagaoka Tech. College, vol. 1, 1962, pp. 11-23.

4. Remarks on the minimum problems of several complex variables, Res. Rep. Nagaoka Tech. College, vol. 1, 1963, pp. 1-6.

MILIN, I. M. 1. Closed orthonormal systems of analytic functions in finitely connected domains, Dokl.

Akad. Nauk SSSR, vol. 157, 1964, pp. 1043-1046. (Russian) 2. The method of area in the theory of univalent functions, Ph.D. dissertation, Leningrad

State University, 1964. 3. The method for schlicht functions in domains of finite connectivity, Trudy Mat. Inst.

Steklov, vol. 94, 1968, pp. 90-121. MlNIATOFF, A.

1. Sur une proprUU des transformations dans I'espace de deux variables complexes, C. R. Acad. Sci. Paris, vol. 200, 1935, pp. 711-713.

2. Zum Interpolationsproblem bei Funktionen mehrerer komplexer Veranderlicher, C. R. Acad. Sci., URSS, nouv. ser., vol. 4, 1935, pp. 243-245.

MITCHELL, J. 1. The kernel function in the geometry of matrices, Duke Math. J., vol. 19,1952, pp. 575-584. 2. Summability methods on matrix spaces, Canad. J. Math., vol. 13, 1961, pp. 63-77. 3. Area and moment of inertia theorems for circular domains in Cn, Duke Math. J., vol.

33, 1966, pp. 209-221. MlTROCHlN, I.

Vber die Veranderung der Krummung von Hyperflachen bei pseudokonformen Abbildungen, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat., vol. 1, 1935-37, pp. 267-280.

MORITA, K. On the kernel functions for symmetric domains, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, vol. 5, 1956, pp. 190-212.

NEHARI, Z. 1. The kernel function and canonical conformal maps, Duke Math. J., vol. 16, 1949,

pp. 165-178. 2. On analytic functions possessing a positive real part, Duke Math. J., vol. 15, 1948,

pp. 1033-1042. 3. The kernel function and the construction of conformal maps, Proc. of a Symposium

on Conformal Mapping, National Bureau of Standards, Appl. Math. Ser., vol. 18, 1952, pp. 215-224.

BIBLIOGRAPHY 247

3a. The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc , vol. 55, 1949, pp. 545-551.

4. On bounded analytic functions, Proc. Amer. Math. Soc , vol. 1, 1950, pp. 268-275. 5. On the numerical computation of mapping functions by orthogonalization, Proc. Nat.

Acad. Sci. U.S.A., vol. 37, 1951, pp. 369-372. 6. Bounded analytic functions, Bull. Amer. Math. Soc , vol. 57, 1951, pp. 354-366. 7. On weighted kernels, J. Analyse Math., vol. 2, 1952, pp. 126-149. 8. An integral equation associated with a function-theoretic extremal problem, J. Analyse

Math., vol. 3, 1954-54, pp. 29-48. 9. On the numerical solution of the Dirichlet problem, Proc. Univ. of Maryland on

Differential Equations, 1956, pp. 157-178. NlRENBERG, R.

On quasi-pseudoconformality in several complex variables, Trans. Amer. Math. Soc , vol. 127, 1967, pp. 233-240.

NORGUET, F. ProbUmes sur les formes difftrentielles et les courants, Ann. Inst. Fourier, Grenoble, vol. 11, 1961, pp. 1-82.

NOWINSKI, J. AND RABINOW1TZ, P. The method of the kernel function in the theory of elastic plates, J. Appl. Math. Phys., vol. 13, 1962, pp. 26-42.

OKA, K. 1. Note sur les families de fonctions analytiques multiformcs etc., J. Sci. Hiroshima Univ.,

Series A, vol. 4, 1934, pp. 94-98. 2. Sur les fonctions analytiques de plusieurs variables, J. Sci. Hiroshima Univ., Series A,

vol. 6, 1936, pp. 245-255; vol. 7, 1937, pp. 115-130; vol. 9, 1939, pp. 8-19. Japan. J. Math., vol. 17, 1941, pp. 517-521; vol. 17, 1941, pp. 523-531. Tohoku Math. J., vol. 49, 1942, pp. 15-52.

3. Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique interieur, J. Math. Soc. Japan, vol. 27, 1953, pp. 97-155.

4. Sur les fonctions analytiques des plusieurs variables, X. Une mode nouvelle engendrant les domaines pseudoconvexes, J. Math. Soc Japan, vol. 32, 1962, pp. 1-12.

OSBORN, H. The Dirichlet functional. I, J. Math. Anal. Apph, vol. 1, 1960, pp. 61-112.

OZAKI, S., KASHIWAGI, S. AND TSUBOI, T. On kernel functions, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A., vol. 5, 1955, pp. 14-19.

OZAKI, S., ONO, J. AND UMEZWA, T. General minimum problems and representative domains, Sci. Rep. Tokyo Daigaku Sect. A, vol. 5, 1955, pp. 1-7.

OZAKI, S. AND TASHIRO, V. Mapping functions onto the representative domain and geodesic lines, Sci. Rep. Tokyo Kyoiku Daigaku Sec A, vol. 10, 1969, pp. 127-133 and 232-248.

OZAWA, M. Fredholm eigen value problem for general domain, Kodai Math. Sem. Rep., vol. 12, 1960, pp. 38-44.

PLEMELJ, J. Ein Erganzungssatz zur Cauchyschen Jntegraldarstellung analytischer Funktionen, Randwerte betreffend, Monatsh. f. Math. u. Phys., vol. 19, 1908, pp. 205-210.

PLESSNER, A. 1. Vber das Verhalten analytischer Funktionen am Rande ihres Definitionsbereiches, J. Reine Angew. Math., vol. 158, 1927, pp. 219-227.

POINCARE, H. 1. Sur les residues des integrates doubles, Acta Math., vol. 9, 1887, pp. 321-380. 2. Les fonctions analytiques de deux variables et la representation conforme, Rend, di

Circolo mat. di Palermo, vol. 23, 1907, pp. 185-220.

248 BIBLIOGRAPHY

POLKINGHORNE, J. C. AND SCREATON, G. R. The analytic properties of perturbation theory, Nuovo Cimento, vol. 15,1960, pp. 289-300.

DE POSSEL, R. Zum Parallelschiltztheorem unendlichvielfach zusammenhangender Gebiete, Gottinger Nachrichten, 1931, pp. 199-202.

PRIVALOFF, J. J. Cauchy's integral, Reports of the Faculty of Physics and Mathematics at the University of Saratow, 1919, pp. 1-96. {Russian)

RADON, J. 1. Theorie und Anwendung der absolut additiuen Mengenfunktionen, Wiener Berichte,

vol. 122, 1913, pp. 1295-1438. 2. Vber lineare FunktionaUransformationen und Funktionalgleichungen, Wiener Berichte,

vol. 128, 1919, pp. 1083-1121. RAMADANOV, I.

Sur une propriete de la fonction de Bergman, C. R. Acad. Bulgare Sci., vol. 20, 1967, pp. 759-762.

REINHARDT, K. Ueber Abbildungen durch analytische Funktionen zweier Veranderlicher, Math . Ann., vol. 83,1921.

RESNJKOFF, H. L. 1. The maximum modulus principle for tubes over domains of positivity, Math . Ann.,

vol. 156, 1964, pp. 340-346. 2. Supplement to "Some remarks on Poincare series", Compositic Math., vol. 21, 1969,

pp. 177-181. RIESZ, F. AND M.

Vber die Randwerte analytischer Funktionen, 4, Congr. Scand. Math. RODIN, B.

1. Reproducing kernels and principal functions, Proc. Amer. Math. Soc., vol. 13, 1962, pp. 982-992.

2. Extremal length of weak homology classes on Riemann surfaces, Proc. Amer. Math. Soc., vol. 15, 1964, pp. 373-374.

ROSENTHAL, P. On the zeros of the Bergman function, Proc. Amer. Math. S o c , vol. 21, 1969, pp. 33-35.

RUDNIK, D. E. 1. On quasiholomorphic mappings, Trudy Moskov. Inst. Electron. Masinostraenija,

vol. 2, 1966, pp. 505-510. (Russian) 2. On quasiholomorphic mappings with a bounded defect in Cn, Trudy Moskov. Inst.

Electron. Masinostraenija, vol. 2, 1966, pp. 505-510. (Russian) 3. On quasiholomorphic mappings of the first rank, Moskow. Oblast. Ped. Inst. Ucen.

Zap., 1967. (Russian) (to appear) RUSEV, P.

1. Analytic transformations which increase the volume and Bergman's function, Izv. Mat . Inst. Bulgar., vol. 8, 1964, pp. 195-201.

2. Analytic mappings and Bergman's function, Doki. Akad. Nauk SSSR, vol. 15, 1964, pp. 279-280. (Russian)

SARIO, L. 1. Existence des fonctions d'allure donnie sur une surface de Riemann arbitraire, C. R.

Acad. Sci. Paris, vol. 229, 1949, pp. 1293-1295. 2. Existence des integrates abeliennes sur les surfaces de Riemann arbitraires, C. R. Acad.

Sci. Paris, vol. 230, 1950, pp. 168-171. SCHIFFER, M .

1. Sur les domaines minima dans la thtorie des transformations pseudo-conformes, C. R. Acad. Sci. Paris, vol. 217, 1938, pp. 112-115.

2. A method of variation within the family of simple functions, Proc. London Math. Soc , vol. 44, 1938, pp. 432-449.

3. Sur la diamMre transfini d'un continu, Bull. Soc. Math. France, vol. 68, 1940, pp. 158-176.

BIBLIOGRAPHY 249

4. Variation of the Green function and theory of the p-valued functions, Amer. J. Math., vol. 65, 1943, pp. 341-369.

5. The span of multiply-connected domains, Duke Math. J., vol. 10, 1943, pp. 209-216. 6. The kernel function of an orthonormal system, Duke Math. J., vol. 13, 1946, pp. 529-540. 7. On the modulus of a doubly-connected domain, Quart. J. Math. (Oxford Ser.), 1946,

pp. 197-213. 8. Hadamard's formula and variation of domain functions, Amer. J. Math., vol. 68, 1946,

pp. 417-448. 9. An application of orthonormal functions in the theory of conformal mapping, Amer.

J. Math., vol. 70, 1948, pp. 147-156. 9a. Faber polynomials in the theory of univalent functions, Bull. Amer. Math. Soc , vol.

54, 1948, pp. 503-517. 10. On various types of orthogonalization, Duke Math. J., vol. 17, 1950. 11. The Fredholm eigenvalues of plane domains, Pacific J. Math., vol. 7,1957, pp. 1187-1225. 12. Applications of variational methods in the theory of conformal mapping, Proc. Symp.

Appl. Math., vol. 8, 1958, pp. 93-113. 13. Fredholm eigenvalues of multiply-connected domains, Pacific J. Math., vol. 9, 1959,

pp. 211-264. 14. Fredholm eigenvalues and conformal mapping, Rend. Mat. e Appl. (3-4), vol. 22, 1963,

pp. 447-468. 15. Half-order differentials on Riemann surfaces, S1AM J. Appl. Math., vol. 14, 1966,

pp. 922-934. SCHIFFER M. AND SPENCER, D . C.

1. Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1954. 2. Coefficient problem for multiply-connected domains, Ann. of Math., vol. 52, 1950,

pp. 352-402. 3. On the conformal mapping of one Riemann surface into another, Ann. Acad. Sci. Fenn.

Ser. Al , vol. 94, 1951, pp. 3-9. SCHMIDT, E.

Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann., vol. 63, 1907, pp. 433-476; vol. 64, 1907, pp. 161-174.

SCHOTTKY, E . Ueber die konforme Abbildung mehrfachzusammenhangender ebener Flachen, J. Reine Angew. Math., vol. 83, 1877.

SCHWARTZ, L. 1. Sous-espaces hilbertiens et anti-noyaux as soc its, Seminaire Bourbaki, vol. 239,

1961-62, pp. 1-18. 2. Spazi di Hilbert e nuclei associati, Centro Internazionale Mate mat ico Estivo, Univ.

de Rome, 1961. 3. Sour-espaces hilbertiens despaces vectoriels topologiques et noyaux associis (noyaux

reproduisants), J. Analyse Math., vol. 13, 1964, pp. 115-256. 4. Sous-espaces hilbertien et noyaux associis; applications aux representations des

groupes de Lie, Colloque d'Analyse Fonctionnelle de Liege (Centre beige de Recherche Mathematiques), 1964, pp. 153-164.

5. Mesures de Radon sur des espaces topologiques arbitraires, Cours mineographe, Institut Poincare, 1964-1965.

SHAPIRO, H.S. Reproducing kernels and Beurlings theorem, Trans. Amer. Math. Soc , vol. 110, 1964, pp. 448-458.

SHAPIRO, H. S. AND SHIELDS, A. L. 1. On the zeros of functions with finite Dirichiet integral and some related function spaces,

Math.Z., vol. 80, 1962, pp. 217-229. 2. Interpolation in Hilbert spaces of analytic functions, Studia Math., vol. 1, 1963,

pp. 109-110. SHIFFMAN, M.

Uniqueness theorems for conformal mapping of multiply connected domains, Proc. Nat. Acad. Sci. U.S.A., vol. 27, 1941, pp. 137-139.

250 BIBLIOGRAPHY

SINGH, V. Grunsky inequalities and coefficients of bounded schlicht functions, Ann. Acad. Sci. Fenn.Ser. A I, No. 310, pp. 1-21.

SKWARCZYNSKI, M. 1. A representation of a bounded function as infinite product in a domain with a Bergman-

Shilov boundary surface, Pacific J. Math., vol. 25, 1968, pp. 177-183. 2. Bergman function and conformal mappings in several complex variables, Bull. Acad.

Polon. Sci. Ser. Sci. Math. Astronom. Phys., vol. 17, No. 3, 1969, pp. 139-140. 3. The invariant distance in the theory of pseudo-conformal transformations and the Lu

Qi-Keng conjecture, Proc. Amer. Math. S o c , vol. 22, 1969, pp. 305-310. 4. The Bergman function and semiconformai mappings (to appear).

SLADKOWSKA, J. 1. Bounds of analytic functions of two complex variables in domains with the Bergman-

Shilov boundary, Pacific J. Math., vol. 12, 1962, pp. 1435-1451. 2. Bounds of analytic functions of two complex variables in domains with the Bergman-

Shilov boundary, Proc. Nat. Acad. Sci. U.S.A., vol. 49, 1963, pp. 217-219. SMATKOV, A. A.

1. Bounds for biholomorphic and quasibiholomorphic transformations of bounded domains, Abstracts of Lectures, Vth AU-Union Conf. Theory of Functions, Erevan, 1960, pp. 124-125. (Russian)

2. Bounds for biholomorphic transformations of bounded domains, lzv. Akad. Nauk Annjan. SSR, vol. 18, 1965, pp. 43-49. (Russian)

3. On a class of quasibiholomorphic transformations, lzv. Akad. Nauk Armjan. SSR, vol. 18, 1965, pp. 26-31. (Russian)

4. On quasibiholomorphic transformations, Abstracts of Lectures, VII th AU-Union Conf. Theory of Functions, Rostov n/Donu, 1963, pp. 188-190. (Russian)

5. On quasibiholomorphic transformations, Moskov. Oblast. Ped. Inst. Ucen. Zap., vol. 137, 1964, pp. 185-192. (Russian)

6. On a class of quasibiholomorphic transformations, Moskov. Oblast. Ped. Inst. Ucen. Zap., vol. 165, 1966, pp. 157-169. (Russian)

SOMMER, F. AND MEHRING, J. Kernfunktion und Hullenbildung in der Funktionentheorie meherer Veranderlichen, Math. Ann., vol. 131, 1956, pp. 1-16.

SPRINGER, G. 1. Pseudo-conformal transformations onto circular domains, Duke Math. J., vol. 18,

1951, pp. 411-424. 2. An extremal problem, Symp. on Complex Analysis (Air Force), Princeton, vol. I,

pp. 340-346. 3. Interpolation problems for functions of several complex variables, J. Math. Mech.,

vol. 7, 1958, pp. 957-962. STARK, J. M.

1. On distortion in pseudo-conformal mapping, Pacific J. Math., vol. 6, 1956, pp. 565-582. 2. Minimum problems in the theory of pseudo-conformal transformations and their

application to estimation of the curvature of the invariant metric, Pacific J. Math., vol. 10, 1960, pp. 1021-1038.

SUFFR1DGE, T . J. On the kernel function for the intersection of two simply connected domains (to appear).

SUGAWARA, M. On the theory of kernel functions, J. Fac. Sci. Univ. Tokyo Sect. I. vol. 7, 1956, pp. 265-304.

SVENSSON, B. E. Y. Relation between the Fourier transform and integral representations for the two and three point functions, Nuclear Phys., vol. 39, 1962, p. 198.

SZEGO, G. Ueber orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehoren, Math. Z., vol. 9, 1921, pp. 218-270.

BIBLIOGRAPHY 251

TASHIRO, Y. 1. On the holomorphic sectional curvature of the Bergman metric, Sci. Rep. Tokyo Kyioku

Daigaku D., Sect. A, vol. 8, 1963, 63-77. 2. The curvature of the Bergman metric, Sci. Rep. Tokyo Kyioku Daigaku Sect. A,

vol. 8, 1965, pp. 280-296. T S U B O I , T .

1. Note on complete elements and kernel functions in several complex variables, Sci. Rep. Tokyo Kyioku Daigaku Sect. A, vol. 5, 1956, pp. 253-286.

2. Notes on Bergman representative domains, Sci. Rep. Saitama Univ., Ser. A3, 1959, pp. 49-64.

3. n-representatie domains and n-minimal domains in several complex variables, Sci. Rep. Saitama Univ., Ser. A, vol. 4, 1961, p. 1-14.

4. Bergman representative domains and minimal domains, Japan. J. Math., vol. 29, 1959. TSUBOI, T. AND MATSUURA, S.

Some canonical domains in Cn and moment of inertia theorems, Duke Math. J., vol. 30, 1969, pp. 517-536.

TSURUMI, K. Potential theory in Bergman spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sec. A, vol. 8, 1963, pp. 122-132.

VAlSALA, J. Quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. IA, 1961, p. 298.

VlNBERC, £ . B. AND GlNDlKlN, S. G. Kahler's manifolds which admit a transitive solvable group of automorphisms, Mat . Sbornik, vol. 74 (116), 1967, pp. 357-377. (Russian)

VLADIMIROV, V. S. 1. Construction of envelopes of holomorphy for regions for a special type and their applica­

tion, Trudy Mat. Inst. Steklov., vol. 60, 1961, pp. 101-144; Engl, transl., Amer. Math.Soc. Transl. (2), vol. 48, 1965, pp. 107-150.

2. A generalization of Liouville's theorem, Trudy Mat. Inst. Steklov., vol. 64, 1961, pp. 9-27; Engl, transl., Amer. Math. Soc. Transl. (2) vol. 43, 1964, 195-213.

3. Functions holomorphic in tubular cones, Izv. Akad. Nauk SSSR Ser. Mat., vol. 27, 1963, pp. 75-100. (Russian)

WALSH, J. L. 1. Determination d'une fonction analytique par ses valeurs donntes dans une infinite

dtnomberable de point, Bull. Soc. Math. Belg. (1954), pp. 52-70. 2. An interpolation problem for harmonic functions, Amer. J . Math. , vol. 76 (1954),

pp. 259-272. WALSH, J. L. AND NILSEN, N.

On functions analytic in a region; approximation in the sense of least p powers, Trans. Amer. Math. Soc , vol. 65, 1959, pp. 239-258.

WARSCHAWSKI, S. Vberdas Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung, Math. Z., vol. 35, 1932, pp. 321-456.

WEIL, A. 1. Sur les series de polynomes de deux variables complexes, C. R. Acad. Sci. Paris, vol.

194, 1932, pp. 1304-1307. 2. L'integrate de Cauchy et les fonctions de plusieurs variables, Math. Ann., vol. I l l , 1935,

pp. 178-182. WEILL, G.

1. Reproducing kernels for analytic differentials on Riemann surfaces, Pacific J . Math., vol. 12, 1960, pp. 729-767.

2. Extremal properties of linear combinations of kernels for analytic differentials on Riemann surfaces, Pacific J. Math., vol. 12, 1962, pp. 1459-1465.

3. Capacity differentials on open Riemann surfaces, Pacific J. Math. , vol. 12, 1962, pp. 769-776.

252 BIBLIOGRAPHY

4. On Bergman's kernel function for some uniformly elliptic partial differential equations, Proc. Amer. Math. Soc., vol. 16, 1965, pp. 1299-1304.

WELKE, H. Uber die analytischen Abbildungen von Kreishbrpern und Hartogsscken Bereichen, Math. Ann., vol. 103, 1930, pp. 437-449.

WlCHTMAN, A. "Analytic functions of several complex variables," in Relations de dispersion et particules elementaires, Ecole d'ete de physique theorique, Les Houches, 1960.

WlRTINGER, W. 1. Zur formalen Theorie der Funktionen von mehreren homplexen Veranderlichen, Math.

Ann., vol. 97, 1927, pp. 357-375. 2. Ueber eine Minimalaufgabe im Geibiet der analytischen Funktionen, Monatsh. f. Math.

u. Phys., vol. 39, 1932, pp. 377-384. WLOKA, J.

Reproduzierende Kerne und nucleare Raume, Math. Ann., vol. 63, 1966, pp. 167-188. WOLF, J. A.

Decomposition of complex manifolds under the action of real groups, Bull. Amer. Math. Soc. (to appear).

WONG, Y. C. Euclidean n -planes in pseudo-euclidean spaces and differential geometry of Carton domains, Bull. Amer. Math. Soc., vol. 75, 1969, pp. 409-414.

YASUDA, J. A note on the kernel function, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, vol. 9, 1965, pp. 74-76.

ZARANKIEWICZ, K. 1. Sur la representation con forme d'un domaine doublement connexe sur un anneau cir-

culaire, C. R. Acad. Sci. Paris, vol. 198, 1934, pp. 1347-1349. 2. Ueber ein numerisches Verfahren zur konformen Abbildung zweifach zusammen-

hangender Gebiete, Zeits. f. ang. Math. u. Mech., vol. 14, 1934, pp. 97-104. ZAREMBA, S.

Sur le calcul numerique des fonctions demandies dans le problime de Dirichlet et le probleme hydrodynamique, Bull. Internat. Acad. Sci., Cracovie, vol. 1,1909, pp. 125-195.

ZYGMUND, A. 1. On the differentiability of multiple integrals, Fund. Math., vol. 23, 1934, pp. 143-149. 2. On the boundary values of functions of several complex variables. I, Fund. Math.,

vol. 36, 1949, pp. 207-235. 3. A remark on functions of several complex variables, Acta. Sci. Math. Szeged, vol. 12,

1950, pp. 66-68.

INDEX A ( r , 5 , / - 1 ) , 222 AU(R),85 Abelian integrals, 49 Ahlfors version of the Schwarz-Pick

lemma, 207 Algol program for solution of a boundary

value problem, 165 Analytic angle, 175 Analyti continuation, 18, £7 Analytic functions of two complex

variables, 166 Analytic hypersurface, 170, 224

with axis reduced to a point (an example), 213

Analytic polyhedron, 168,171 Analytic surface, 170 Angle of deviation, 175 Approach A to a boundary point, 204 Area A of the complement B of B

(inequalities for A), 78 Area AU(R) of R with respect to u <£ Rt 85 Average quadratic error, 3 Axial symmetric harmonic functions, 165 Axis of representation of a segment of an

analytic hypersurface, 211

B-area of a surface, 176 Basic space, 33 Behavior on the boundary of the metric

and the curvature, 38, 39, 204 Bessel inequality, 3 Bicylinder, 167 Biharmonic function (s), 214, 226 Bjorken-Gilbert-Landau conditions, 151 Bound for the growth of a function, 20 Bound for the lowest eigenvalue \\ of

equation V, (97), 78 Boundary behavior of the kernel function,

38,203 Boundary conditions for a plate

with clamped edges, 163 with free edges, 163

Boundary of four-dimensional domain, 167 Boundary value problem of first and

second kinds, 144 Bounded functions, 117

Canonical domains, 89 Canonical mapping functions, in terms of

kernel function, 99

Cauchy formula (generalized), 173,174 Cauchy sequence, 43 Cauchy-Riemann equations, 48

(generalized), 163 in the theory of functions of two complex

variables, 166,180 Chern class, 186 Christoffel symbols, 184,185 Circular domain, 168

C, 168,190 Classical domain functions, 46 Closed orthogonal system, existence of, 10

for a simply-connected domain, 12 Closed system of orthogonal functions, 3 ,10 Compact family, 10,15 Complement B of B, 78 Complete space, 43 Completely additive set function, 227 Complex derivatives, 35 Conformal capacity (introduced by

Loewner), 210 Conjugate harmonic function, 48 Connection between the kernel and

Green's function, 60 Countable basis, 43 Curvature

in analytic direction {ua) , 185 bounds for, 36,37 Gaussian, 32 of invariant metric, 35 radius of, 32 of a surface, 175 total, 32

Curve, length of, 31

£ ( * , * ) , 47 Darboux sums, 132 A„(z,f),83 Determinantal notation, 25 Diffeomorphism, 206 Differential equations, of elliptic type, 144

with a non-definite coefficient, 155 Differential geometry, 31 Differential operators, 35 Differential parameter of Schwarz {<t>, z), 64 Dirichlet integral, 46,147

as metric, 50 Dirichlet problem, 54,144

solved using kernel function, 54

253

254 INDEX

Distinguished boundary, 172 D2(r),221,224

Distortion of length, 206 bounds for, 206

Distortion (maximal) in quasi-conformal mapping, 210

Distortion theory, 39 Domain, canonical, 89

for which polynomials are closed, 13 Domain (s) of comparison, 37,199

examples, 202 Domain functions, 46

monotonicity of, 127 Doubly harmonic functions, 214 Doubly orthogonal functions, 14

E(\, H, a) (approach E(\, <fi2, <*)), 226 Eigenvalues and eigenfiinctions of LB(z, f), 72 e-isometry, 209 Elasticity equation, AA<*> = 0,160

solution of boundary value problems for, 163

Element of the B-area, <fft(2)(zlfz2),201 Elliptic differential equations, 42,144 Entire functions, 221 e -quasi-conformal, 210 Equation of elasticity, AA^ = 0,160 Equation between Green's function and

kernel function, 61,62,162 Equivalence class, 33 Euler-Lagrange equation, 145 Exceptional values for the hypersurface s and the function /, 222

Existence of slit mappings, 143 Expansion theorems, 7,145,149,163,178 Extended class of functions, 214 Exterior domain of comparison, 200 Exterior normal, 226 Extremal curves, 31 Extremal properties of canonical domains, 95

F,(z),48 Fatou's theorem (generalized), 224 Feyroan's integral, 151 First boundary value problem, 54 First fundamental form, 31 Four dimansional domains, 166

visualization of, 167 Fourier coefficient, 2 Fourier expansion, 148 Fubini-Study metric, 186 Functions

compact family of, 10 doubly-harmonic, 214

doubly-orthogonal, 14 equi-continuous family of, 145 extended class of, 228 of extended class (complex), 215 of extended class (real), 214, 217 of the first kind, 49 of upper extended class, 228 of lower extended class, 228 orthogonal with respect to the distinguished

boundary surface, 229 of the second kind, 49 of third kind, 49 of two complex variables, 166,176

Fundamental domain, 43 Fundamental solution S(Z, Z), 144

G(2,f ) ,52,60,62 8(z, fK83 rfl("U-),65 TB(z, fK 80 r(p,(z,rt,83 Gaussian curvature, 32 Genus of a closed Riemann surface, 49 Geodesies, 31 Geometric quantities, 65 Gram-Schmidt orthogonalization procedure, 3 Green's formula, 47 Green's function, 46

construction of, 151 eigenfiinctions, 154 of extended class, 215 relation to Neumann's function, 52 representation, 52,153

by kernel function, 56 variation of, 132

Grunsky necessary and sufficient conditions for uni valency, 87

Grunsky inequalities, 202

Hp (class Hp), 225 Had am aid variational method, 122,124 Harmonic functions, 46

closed system of, 49 Harmonic measure, 47

conjugate harmonic function, 48 representation in terms of kernel

function, 55 Hermitian bilinear form, 28 Hermitian form, 18 Hermitian metric, 184 Hermitian symmetric space, 206 Hilbert space, 42, 43

with kernel function, 45 Homeomorphism, 206

INDEX 255

Hyperbolic distance p(ft, ft) in the theory of one variable, 198

Hyperbolic measure, generalization of, 199 Hyperbolic metric, 35 Hypersphere, 167

Identities connecting domain functions, 62 Indicatrix of the metric, 196 Indicatrix of a Riemannian manifold (in the

space of one complex variable), 39 Infinitesimal PCT, 186 Integral operator of first kind, 149 Integrals of first kind, 49 Interior distinguished set, 41,194 Interior domain of comparison, 200 Interior normal, 227 Invariant metric, 32, 33,183 Invariant with respect to PCT's, 183

«/</,*), 1 JB(ZU ZT) invariant, 198 Jacobi reduction of a form, 28 JumpA(T/(z))ofT/(z) ,67

*(z, ^ 5 2 # B ( * , £ ) , 2 9 , 6 2 , 1 7 8 KB(2,£) ,49,80 KBiz.p, 110 /tx(z,r),(A = A(z )>0) ,120 ko(z, 0 ,122 M * . £ ) , 5 9 K(z, rt, 110 X n (z , f ) ,108 K,(z,r),82_ K(zltz2ttlft2), 229 Kahler manifold with constant

curvature R, 187 Kahler metric, 184 Kernel of Cauchy formula, 174 Kernel of a system of orthonormal

functions, 8,178 Kernel function, 43,61,80,108,138,147,

148,153,159,160,163 boundary behavior of, 38 boundary relation of, 143 canonical mapping functions in terms of, 99 for circle, 9,110 of the class/ ,49 extremal property, 118,180 harmonic, 51 identities, 59 independence of a closed system of

orthogonal functions, 21,178

inequality in terms of, 23 invariance of, 180 K(zJ) of analytic functions, 21 minimum property of, 44 monotonicity of, 44 for non-bounded, non-schlict domains, 37 of an orthogonal system, 9 relative invariant, 33 reproducing property of, 22,44,114 for ring, 10 for simply-connected domains, 13,110 ofspaceA,(B),80 in two complex variables, 178 uniqueness of, 23 variation of, 122

/ ,63 X2 = X2(B) (see also footnote 6, p. 67), 1 / 2 (B) ,49 £jj(*, r),80 LB(z,rt, 62,63 X 2 (B) ,49 X2(B), Xoio(B), X&,i(J3) (classes of

functions), 191 IB(W> ") series development, 77 lB{w,J) transformation law OHB(W,U) in

conformal mapping, 77 L (D ) (D the distinguished boundary surface), 229

L ^ (Hermitian form), 202 L (kernel L), 62 Lp (class Lp), 225 /«(*.«. 82 Lt(z,t)f 82 Ht„ %)t inequalities for, 66 L(z,t), 110, 111 Lamina of a segment of an analytic

hypersurface, 172 A2(B),47 /&(B), 122 A2(B),50 4(0 , 21 Xjs (zi,z2), 199 Ar (space), 69 X, (eigenvalues), 72 A, (space A«), 74, 80, 82 Laplace equation, 144,145 Length, an absolute invariant, 33

of a curve, 31 Level hypersurfaces J (z\, z2, zi,z2) = const, of the invariant J for the domains l * i | 2 / P + l * 2 l 2 < 1.195

Level lines J (z,z) = const., 40,41

256 INDEX

Limit relations when approaching a boundary point, 204, 205

Linear independence, 1 Linear space, 43 Loewner's conformal capacity, 210 Logarithmic singularity, (singularities),

46,152 L-transform of/, T/(z), 67 /•transformation, 67

Mapping function, for simply-connected domain, 24

existence of, 98 Mapping radius d of a domain (inequalities

ford), 78 Mapping theorems, 89 Matrix notation, 24 Maximum boundary set, 168 M i ( z , 0 , 2 1

Meromorphic functions, 221 of several complex variables, 221, 223

Method of minimum integral, 39,198 Method of orthogonal functions in the theory

of analytic functions, 176 Metric, invariant with respect to CT, 32

boundary behavior of, 37, 38 Gaussian curvature of, 38, 39 monotonicity of, 34 of Pick-Poincare, 34, 35

Metric invariant with respect to PCT's, 182,183

Metric on a complex manifold, 185 Minimal domain with respect to t, 24 Minimal domain with respect to (*i, t2), 193 Minimal problems, 178 Minimizing function, 21 Minimum area, 24 Minimum functions, 181 Minimum problem, 24, 30

and partial differential equations, 145 Minimum problems with interpolation conditions, 30

Minimum value XXQ0'"Xmn(t), 178,179,198 M-minimal domain, 24 Modulus of a doubly-connected domain, 134 Moments of the Xth degree, 222, 223 Monotonic properties of minimum X's, 36,199 M ( r , / ) , 221 Multiply-connected domains, 89

N (condition JV), 225 N (condition N ) , 225 N(2 , r t , 52 ,61 ,62 N(2;f ,*b),53

n-dimensional complex projective space Pn, 187

n-dimensional euclidean space C", 187 Nehari's theorem about kx(z, 7), 120 Neumann's function, 46,144,148

construction of, 151 relation to Green's function, 52 representation of, 52,153

Neumann's problem, 55,144 solved with the help of kernel function, 55

Nevanlinna's formula (generalized), 219 Nevanlinna's theory of meromorphic func­

tions of one complex variable, 216 N on -Euclidean length, 34 Non-Euclidean volume element, dil (z l f z2),

200 Norm, in Hilbert space, 43 Normal derivative, 144 Normal family, 135, 211 Normal manifold, 185

holomorphic sectional curvature, 185 unitary curvature, 185

Normalization, 2 Normed space, 43

O(M), extended class O(M), 217 fl2,43 «,(z),47 Orbit, 169 Orthogonal functions, 2

of two complex variables, 176 Orthogonal harmonic functions, 49 Orthogonal trajectories, 41 Orthogonalization over the boundary, 108 Orthogonalization process, 3 Orthonormal functions, 2 Orthonormal system (s),

for elliptic differential equations, 151 for equation AA4 = 0,161 of functions, 2

Orthonormalization conditions, 177

Parseval formula, 3 Parseval identity, 152 Partial differential equation (s), 144

fundamental solution of a, 144 of elliptic type, 144

PCT (pseudo-conformal transformations), 166 normalized, 188 onto the representative with distinguished

point (*i,*2),188,189 P(z ,r ) ,48 P,„, period of F,{z), 48 K ( z ) | , 8 1 #„(z) (eigenfunctions), 75

INDEX 257

4>(z,u), 82 Plemelj's theorem, 67,104 Plessner's theorem, 228 Poisson formula (generalized), 176, 215 Poisson's ratio, 162,164 Poisson-Jensen formula (generalized), 218 Polygonal domains, 4 Polyhedron (analytic), 168,171, 224 Positive definite form, 28 Positive definite Hermitian quadratic form, 44 Positive definite Hermitian scalar product, 43 Potential-theoretical method, 228 Potential theoretical methods in the theory

of functions of two complex variables, 211 Principle, of hyperbolic measure, 34

of minimum integral, 39,178 Principal minors of the determinant £&! .x - X.£*«x. 202

Privaloff's theorem, 104 Projection into the class ft , 154 Pseudo-conformal equivalence, 187,188 Pseudo-conformal isometry, 186 Pseudo-conformal transformation (PCT),

166,182 of a circular domain, 181,197 invariant of, 183

^Mp) (eigenfunctions), 72 *,p (eigenfunctions), 72 *,(*) = i ( l - A , - 2 r l / 2 T V , ( 2 ) , 7 9 *(z ,u) , 82 P(z;u,v),80

Quasi-isometry, 209 Qfcu f i / ) ,80

Regularity conditions in terms of (as \ for a function 4>{z) = ^aszs in a domain D, 18, 87

Reinhardt circular domain, 169, 194 Representative coordinates at *, 189 Representative (representative domain), 187 Representative domains, 105 Representative R(B, t), 105,188 Representative with a distinguished point, 188 Reproducing property, 21 Ricci curvature, 184 Ricci tensor, 184 Riemann (sectional) curvature, 184 Rieraann curvature tensor, 184 Riemann mapping theorem, 34 Riemannian manifold, 31 Riesz-Fischer theorem, 5 Robin's constant, 128 Robin's problem, 164 Runge's theorem, 13

S 3 = U r D 2 (r) ,221 Scalar product, 1 Schiffer variational method, 130 Schottky theory (see footnote 3), 49 Schwarz inequality, 1 Schwarz's lemma, 44

(generalized), 45 Schwarz-Christoffel constants, 103 Schwarz-Pick (lemma of), 198, 206, 207 Second boundary value problem, 55 Sectorial approach, 226 Sectorial limit, 227 Simply connected domain, 34 Slit mapping, 98,102 Smooth curves, 46 Space

linear, normed, complete, 43 Spherical harmonics, 165 Stieltjes integral, 132 Strong derivative, 227 Surface, 31 Symmetry of G and N, 46 Symmetry relation for periods p,„ of

normalized functions of the first kind, 48 Szego kernel function, 110

(generalized), 229

T/(z),67 TK(z,w)>68 T>„ orthogonality of (T>„), v = 1, 2, • • •, 79 T[T/(z)],69 Totally geodesic analytic surface, 191 Triangle inequality in Hilbert space, 43 Triply connected domain (in the plane), 40, 41 Tubular domain, 169,170 Two-dimensional harmonic functions, 49, 165

Univalency (necessary and sufficient conditions), 87

Univalent function, 86 Universal covering surface, 35

Variation, of analytic kernel, 126 of Green's function, 125,132 of harmonic measures, 126 of kernel function, 124 of period p ^ , 127

Variational method, of Hadamard, 122 of Schiffer, 130

Weierstrass approximation theorem, 151 Weierstrass p-function, 10

f-function, 10 Weighted kernels, 120 Weighting functions, 20