Entire Functions of Several Complex Variables, Softcover reprint of the original 1st ed. 1986 Grundlehren der mathematischen Wissenschaften Series, Vol. 282

I - Entire functions of several complex variables constitute an important and original chapter in complex analysis. The study is often motivated by certain applications to specific problems in other areas of mathematics: partial differential equations via the Fourier-Laplace transformation and convolution operators, analytic number theory and problems of transcen­ dence, or approximation theory, just to name a few. What is important for these applications is to find solutions which satisfy certain growth conditions. The specific problem defines inherently a growth scale, and one seeks a solution of the problem which satisfies certain growth conditions on this scale, and sometimes solutions of minimal asymp­ totic growth or optimal solutions in some sense. For one complex variable the study of solutions with growth conditions forms the core of the classical theory of entire functions and, historically, the relationship between the number of zeros of an entire function f(z) of one complex variable and the growth of If I (or equivalently log If I) was the first example of a systematic study of growth conditions in a general setting. Problems with growth conditions on the solutions demand much more precise information than existence theorems. The correspondence between two scales of growth can be interpreted often as a correspondence between families of bounded sets in certain Frechet spaces. However, for applications it is of utmost importance to develop precise and explicit representations of the solutions.

1. Measures of Growth.- §1. Preliminaries.- § 2. Subharmonic and Plurisubharmonic Functions.- §3. Norms on ?n and Order of Growth.- §4. Minimal Growth: Liouville’s Theorem and Generalizations.- § 5. Entire Functions of Finite Order.- §6. Proximate Orders.- §7. Regularizations.- § 8. Indicator of Growth Functions.- § 9. Exceptional Sets for Growth Conditions.- Historical Notes.- 2. Local Metric Properties of Zero Sets and Positive Closed Currents.- §1. Positive Currents.- §2. Exterior Product.- §3. Positive Closed Currents.- § 4. Positive Closed Currents of Degree 1.- §5. Analytic Varieties and Currents of Integration.- Historical Notes.- 3. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set.- §1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor.- § 2. Indicators of Growth of Cousin Data in ?n.- §3. Canonical Potentials in ?m.- § 4. The Canonical Representation of Entire Functions of Finite Order.- §5. Solution of the ? $$\bar \partial$$ Equation.- § 6. The Case of a Cousin Data.- §7. Slowly Increasing Cousin Data: the Genus q = 0; the Algebraic Case.- §8. The Case of Integral Order: Extension of a Theorem of Lindelöf.- § 9. Trace of a Cousin Data on Complex Lines.- § 10. The Case of a Cousin Data of Infinite Order.- Historical Notes.- 4. Functions of Regular Growth.- § 1. General Properties of Functions of Regular growth.- §2. Distribution of the Zeros of Functions of Regular Growth.- Historical Notes.- 5. Holomorphic Mappings from ?n to ?m.- §1. Representation of an Analytic Variety Y in ?n as F-1(0).- §2. Local Potentials and the Defect of Plurisubharmonicity.- §3. Global Potentials.- § 4. Construction of a System F of Entire Functions such that Y=F-1(0).- §5. The Case of Slow Growth.- §6. The Algebraic Case.- §7. The Pseudo Algebraic Case.- §8. Counterexamples to Uniform Upper Bounds.- §9. An Upper Bound for the Area of F-1(a) for a Holomorphic Map.- § 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes.- Historical Notes.- 6. Application of Entire Functions in Number Theory.- §1. Preliminaries from Number Theory.- §2. A Schwarz Lemma.- §3. Statement and Proof of the Main Theorem.- Historical Notes.- 7. The Indicator of Growth Theorem.- Historical Notes.- 8. Analytic Functionals.- §1. Convex Sets and the Fourier-Borel Transform.- §2. The Projective Indicator.- §3. The Projective Laplace Transform.- §4. The Case of M a Complex Submanifold of ?n.- §5. The Generalized Laplace Transform and Indicator Function.- §6. Support for Analytic Functionals.- §7. Unique Supports for Domains in ?n.- §8. Unique Convex Supports.- Historical Notes.- 9. Convolution Operators on Linear Spaces of Entire Functions.- §1. Linear Topological Spaces of Entire Functions.- § 2. Theorems of Division.- §3. Applications of Convolution Operators in the Spaces Ep?(r)and Eo.- §4. Supplementary Results for Proximate Orders with ?>1.- §5. The Case ?=1.- § 6. More on Functions of Order Less than One.- § 7. Convolution Operators in ?n.- Historical Notes.- Appendix I. Subharmonic and Plurisubharmonic Functions.- Appendix II. The Existence of Proximate Orders.