The Asymptotic Expansion of Integral Functions of Finite Non-Zero Order

E. W. Barnes
1905 Proceedings of the London Mathematical Society  
1. In three memoirs* which have been recently published I have investigated the asymptotic expansions of the logarithms of integral functions of finite order, and suggested that such investigations may be regarded as preliminary to a classification of integral functions. The expansions were obtained for functions of simple and multiple! linear sequence, and it was shewn that expansions for similar functions with certain types of repeated sequence could be deduced: such deductions were made in
more » ... ions were made in certain cases. The investigation was based entirely on the theory of divergent series : in the first memoir I attempted to develop the theory of Borel for this purpose. Throughout the invet igation no attempt was made to determine remainders for the asymptotic expansions. The fundamental procedure consisted in applying the asymptotic expansions of the Maclaurin sumformula to a transformation by logarithmic expansion of the function investigated. The terms of the double series which arose in this way were rearranged, and were then summed by an application of Fourier's series. In order to make the application, it was assumed that | z \, when | z | is large, was of a limited type of number, and a further assumption was made that this limitation could not affect the validity of the result; that, in fact, the form of the asymptotic expansion did not depend on the arithmetic nature of \z\. This assumption is valid in the case of functions of finite (non-zero) order. It seems, however, advisable to undertake the investigation from another point of view. The theory of divergent series is but little known:
doi:10.1112/plms/s2-3.1.273 fatcat:zaqonayuhrbhdkmijqc4zl3tte