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Integral functions. (English) Zbl 0075.05901
Cambridge Tracts in Mathematics and Mathematical Physics, No. 44. Cambridge: At the University Press. viii, 135 pp. (1956).
This useful book deals primarily with the theory of analytic functions (whether entire or not) which are of finite order in an angle. Only the principal results are given, but even these have never before been collected in one place. Little account is taken of developments since about 1940, but most of the theory dealt with here was already in final form by that time. The first two chapters are introductory, dealing with such matters as various versions of the Poisson-Jensen formula and Carleman’s formula, and with generalities about entire functions of finite order. The third chapter is a detailed account of Phragmén-Lindelöf theorems and the Phragmén-Lindelöf function \(h(\theta)\). Chapter 4 gives a detailed account of Lindelöf proximate orders and shows in particular how they make the classical minimum modulus theorems almost obvious. The next two chapters deal in detail, and in the full generality that comes from using proximate orders, with the (far from obvious) relationships between the growth of a function and the distribution of its zeros. Chapter 7 gives the main theorems on Julia lines. The final chapter discusses entire functions of exponential type, the indicator diagram and the Borel polygon, and culminates in theorems connecting the singular points of the Borel-Laplace transform of an entire function with the exceptional values of the function itself.

30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
30Dxx Entire and meromorphic functions of one complex variable, and related topics