(Abstract.)
The memoir deals with the asymptotic expansion of functions with a single essential singularity at infinity in the neighbourhood of that singularity. The term "integral function" is used as a translation of the French expression "fonction entiere."
Part I opens with an introduction, in which it is pointed out that for each of the few integral functions whose detailed properties have been investigated, there always exists near infinity an asymptotic expansion valid in those parts of the region near infinity which are not at a finite distance from zeros of the function. It is then suggested that wide classes of integral functions admit such expansions ; and it is pointed out that, if such a theorem can be proved and the expansions obtained, we may solve many questions relating to the " genre " of a function, to the nature of its zeros, and to the character of its coefficients when expanded as a Taylor's series. We may, in fact, to a large extent classify such functions by their behaviour at infinity.
After the introduction, a short account of the historical development of the enquiry is given ; and then the memoir proper commences with a formal arrangement of integral functions according to the laws of distribution of the zeros as seen in expressions in Weierstrassian- product form.
When the nth zero a n is such that it depends solely upon n and certain definite constants, and also such that the law of dependence is the same for all zeros, we call it a simple integral function with a single sequence of non-repeated zeros, or sometimes briefly a simple integral function.
The zeros are said to be algebraic, when they are given, when n is large, by a formula of the form
fl + -^- + -=- + . . .1 , ?iPi HP*
where p is a rational positive quantity, and pi, p, . . . are rational positive quantities arranged in ascending order of magnitude. If p
is such that 2 . - converges, and 1' , : diverges, however
i I fl P + e i fl P e
n=l | "ft | "=1 1 ' n I
small the positive quantity may be, the function is said to be of order p.
We can form a scale of simple integral functions arranged according
