Abstract
IN elementary algebra the well-known remainder theorem enables us to determine a polynomial, except for a numerical factor, when all the zeroes are given. If we replace the polynomial by an integral function with an infinite number of zeroes, we can still determine a good deal about the function, though not so much as about the polynomial. In 1879 Picard proved a theorem which at that time appeared to have no connexion with the preceding results. He showed that a function that is uniform in the vicinity of an isolated essental singularity takes infinitely many times every value with the possible exception of two. Much later Borel and others linked up these two subjects of investigation, and studied the distribution of values of a complex variable for which a meromorphic function is equal to a given constant. This is the principal topic dealt with by the book under review. The discussion is based on the Poisson-Jensen formula, which connects the modulus of a meromorphic function at any point within a circle with its values on the circumference and the position of its zeroes and poles inside. For lack of space, Prof. Nevanlinna confines himself to a consideration of the moduli. The other half of the problem, the discussion of the arguments of the roots, can be found in Valiron's “Lectures on the General Theory of Integral Functions”(Toulouse, 1923) and elsewhere.
Le théorème de Picard-Borel et la théorie des fonctions méromorphes.
Par Prof. Rolf Nevanlinna. (Collection de monographies sur la théorie des fonctions.) Pp. vii + 174. (Paris: Gauthier-Villars et Cie, 1929.) 35 francs.
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P., H. Mathematical and Physical Sciences. Nature 124, 542 (1929). https://doi.org/10.1038/124542b0
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DOI: https://doi.org/10.1038/124542b0