ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR RESEARCH ON FUNCTIONS OF A COMPLEX VARIABLE WILFRED KAPLAN Associate Professor of Mathematics Project 2150 GRANTS NSF-2286 and NSF-G779 NATIONAL SCIENCE FOUNDATION WASHINGTON, D. C. November 1954

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT The research program on functions of a complex variable has so far resulted in three papers: (1) Curve-families and Riemann surfaces, to be published shortly; (2) Extensions of the Gross Star Theorem, which appeared in the Michigan Mathematical Journal; and (3) Approximation by entire functions, which will be submitted for publication in the near future. Work has also been carried out on extending the Phragmen-LindelSf principle by relating it to growth properties of entire functions. ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN RESEARCH ON FUNCTIONS OF A COMPLEX VARIABLE INTRODUCTION Two grants have been awarded by the National Science Foundation to the University of Michigan for research on functions of a complex variable, the first one (NSF-2286) covering the period from July 1, 1953, to July 1, 1954, and the second (NSF-G779) covering the period from July 1, 1954, to September 1, 1954, In both cases the research was to be carried out by Wilfred Kaplan, Associate Professor of Mathematics. Professor Kaplan was on sabbatical leave for the academic year 1955-1954 and was able to devote the entire period from July 1, 19535 to September 1, 1954, to the research. In August, 19535 he left for Europe for a three-month visit, during which he conferred with mathematicians in England, Switzerland, Finland, Sweden, and Denmark on the subject of the research. Except for this period and brief vacations, the work was done in Ann Arbor. RESEARCH COMPLETED The research program was led to three separate papers which are in various stages of completion as described below: 1. Curve-Families and Riemann Surfaces A new proof is given of the theorem: For every abstract normal chordal system E there exists a function u.(x,y) harmnic for + y2 < 1 or for x 7 + y4 < m such that the family of level curves of u forms a chordal system isomorphic to E. The proof uses only the simplest properties of regular curve-families, and the uniformigation theorem of function theory. ---- 1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Accordingly, it leads to a considerable simplification of the theory of cure -families. This paper was announced at the Conference on Functions of a Complex Variable in Ann Arbor in June, 1953. It was written during the following months and will appear shortly in the proceedings of that conference. 2. Extensions of the Gross Star Theorem Several theormes of the type of the Gross Star Theorem are proved. The following is typical: Let w = 0(z) be meromorphic for |zl < 1. Let B be a closed subset of Izl = 1 having capacity zero. Then each element of the inverse 0l (w) of,(z) can be continued indefinitely in almost all directions for which z = -l(w) approaches a point of B. The concept of "all directions" includes more general families of paths than rays. This paper has appeared in the Michigan Mathematical Journal [vol 2 (1953-1954), PP 105-108]. 3. Approximation by Entire Functions In 1927 Carleman extended the classical Weierstrass approximation theroem to approximation of continuous functions f(x) on the interval [,oooo] by entire functions. This result is amplified and applied in several ways. It is shown that if f(x) has a continuous derivative, then the approximating function 0(z) can be chosen so that t'(x) also approximates f(x). It is shown that, if f(z) is analytic in a domain D bounded by a single open curve C extending to infinity in both directions, and if f is sufficiently smooth on C, then f can be approximated uniformly in D by an entire function. Existence of solutions of the Dirichlet problem for the unit circle for a large class of nonintegrable boundary functions is established. This paper requires some minor editing and will shortly be submitted to the Michigan Mathematical Journal. RESEARCH IN PROGRESS In addition to the work described, a considerable amount of time was devoted to a program of extending the Phragmen-Lindelof principle by relating it to growth properties of entire functions. Some significant results were obtained, but more work is needed to round out the program and give a satisfactory general theory. I —------------ 2

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