Open AccessV. The asymptotic expansion of integral functions defined by Taylor's series
Author(s) -
E. W. Barnes
Publication year - 1906
Publication title -
philosophical transactions of the royal society of london. series a, containing papers of a mathematical or physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9258
pISSN - 0264-3952
DOI - 10.1098/rsta.1906.0019
Subject(s) - taylor series , mathematics , simple (philosophy) , series (stratigraphy) , singularity , infinity , asymptotic expansion , series expansion , transcendental number , function (biology) , order (exchange) , mathematical analysis , essential singularity , transcendental function , calculus (dental) , pure mathematics , medicine , paleontology , philosophy , epistemology , finance , dentistry , evolutionary biology , economics , biology
1. Integral functions can be defined either by Taylor’s series or Weierstrassian products. When the zeros are simple functions of their order number, the latter method is, as a rule, most simple. When the zeros, however, are transcendental functions of the order number, those integral functions which so far have occurred in analysis have been defined by Taylor’s series. [Definitions by definite integrals have usually been reducible to one of the preceding forms.] Whatever be the manner of its definition, an integral function has a single essential singularity at infinity, and the behaviour near this singularity serves to classify the function. By studying this behaviour we may hope to find connecting links between the two modes of definition.