V. The asymptotic expansion of integral functions defined by Taylor's series

Abstract

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.