Abstract
THIS appears as the third instalment of Prof. Borel's lectures on the theory of functions. It is somewhat more fragmentary than its predecessors, and has, in fact, the typical qualities and defects of a set of lecture-notes. As an introduction to the memoirs of Hadamard, Mittag-Leffler and Poincaré, as well as to those of Prof. Borel himself, these chapters will be very serviceable. Perhaps the most noteworthy articles are those which deal with the theory of increment (croissance); it is there shown that there is no natural scale of orders of magnitude. In fact, an aggregate of orders of increasing functions can be constructed which is not numerable. Moreover, ! functions have been invented which have no regular order of increase; thus an example is given of a function j which is comparable with exp x for an infinite number of values of the variable, and with exp (exp x) for another infinite number of values. This will cause searchings of; heart in certain quarters, no doubt; even Prof. Borel remarks that “fort heureusement, les fonctions qui se présentent naturellement aux géomètres sont, en général, de nature plus simple.”
Leçons sur les Séries à termes positifs.
Par Èmile Borel. Recueillies et rédigées par Robert d'Adhémar. Pp. viii + 94. (Paris: Gauthier-Villars, 1902.) Price fr. 3.50.
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Leçons sur les Séries à termes positifs . Nature 66, 5 (1902). https://doi.org/10.1038/066005b0
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DOI: https://doi.org/10.1038/066005b0