native settlement, in 1950 he graduated - as an extramural studen- from Groznyi Teachers College and in 1957 from Rostov University. He taught mathematics in Novocherkask Polytechnic Institute and its branch in the town of Shachty. That was when his mathematical talent blossomed and he obtained the main results given in the present monograph. In 1969 N. V. Govorov received the degree of Doctor of Mathematics and the aca demic rank of a Professor. From 1970 until his tragic death on 24 April 1981, N. V. Govorov worked as Head of the Department of Mathematical Anal ysis of Kuban' University actively engaged in preparing new courses and teaching young mathematicians. His original mathematical talent, vivid reactions, kindness bordering on self-sacrifice made him highly respected by everybody who knew him. In preparing this book for publication I was given substantial assistance by E. D. Fainberg and A. I. Heifiz, while V. M. Govorova took a significant part of the technical work with the manuscript. Professor C. Prather con tributed substantial assistance in preparing the English translation of the book. I. V. Ostrovskii. PREFACE The classic statement of the Riemann boundary problem consists in finding a function (z) which is analytic and bounded in two domains D+ and D-, with a common boundary - a smooth closed contour L admitting a continuous extension onto L both from D+ and D- and satisfying on L the boundary condition +(t) = G(t)-(t) + g(t).
Table of ContentsI.- General Properties of Analytic and Finite Order Functions in the Half-Plane.- 1 Definition of order and indicator of a function holomorphic in an angle. Relations between various definitions of order..- 2 Generalized Nevanlinna and Carleman formulas.- 3 Canonical representation of a function of finite order in the half-plane.- Necessary Conditions of Completely Regular Growth in the Half-Plane.- 4 Definition of completely regular growth in the half-plane. List of results on completely regular growth.- 5 Relation between completely regular growth in open and closed angles.- 6 Asymptotic behavior of the modulus and zero distributions of entire functions of the class A*?.- 7 Existence of argument boundary density for the zero set of a function from the class A*?.- 8 Existence of boundary and argument densities for the zero set of a function from A*?.- Sufficient Conditions of Completely Regular Growth in The Half-Plane and Formulas For Indicators.- 9 The growth of some auxiliary functions of non-integer order.- 10 A criterion for a function to belong to the class A*?, ? being non-integer.- 11 A criterion for a function to belong to the class ¯A*?, ? being non-integer.- 12 The argument-boundary symmetry of the zero set of a function of the class A*?, ? being integer.- 13 The growth of some auxiliary functions of integer order.- 14 A criterion for a function to belong to the class A*?, ? integer.- 15 A criterion for a function to belong to the class ¯A*?, ? integer.- 16 Functions of the class ¯A*? for even and for odd ?.- 17 Functions of a finite degree in the half-plane.- II.- Riemann Boundary Problem With an Infinite Index When the Verticity Index is Less Than 1/2.- 18 Statement of the homogeneous problem.- 19 Canonical function.- 20 Solution of the homogeneous problem in the class BL. Description of solutions of order ?.- 21 Formulation of the non-homogeneous problem and an approach to its solution.- 22 Solution of the non-homogeneous problem.- Riemann Boundary Problem With Infinite Index in The Case Of Verticity of Infinite Order.- 23 Statement of the homogeneous problem.- 24 Canonical function.- 25 Asymptotic properties of zero sets of solutions of the homogeneous problem from the classes B and B*?.- 26 General form of solutions of the homogeneous problem in the class B.- 27 General form of solutions of the homogeneous problem in the class B*?.- 28 An example of a solution of the homogeneous problem in the class B?. Importance of the restriction on the exponent in the Hölder condition for the function ?(t) = arg G(t)/(2?t?).- 29 Statement of the non-homogeneous problem and an approach to its solution.- 30 Auxiliary statements.- 31 Solution of the homogeneous problem.- Riemann Boundary Problem With A Negative Index.- 32 An example of a solvable homogeneous problem with a negative index.- 33 Conditions of unsolvability of the homogeneous problem with a negative index.- 34 Conditions of solvability of the non-homogeneous problem with an index ?.- On the Paley Problem.- A.1 Formulation of the problem and proff of the main inequality.- A.2 Solution of the Paley problem.