MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
449 
For unless J (2) and the series a {) + j + . . . + have such a character, it 
is impossible that L t 
J (2) - « 0 “ " 
" should, as 2 approaches infinity, 
always tend uniformly to zero, whatever be the argument of 2. 
But, as we proceed to show, a uniform function of finite genre,” with an essential 
singularity at infinity can, in general, be represented by one or more asymptotic 
expansions valid for all points near infinity except those in the immediate vicinity of 
the zeros and poles of the function. 
Two dilferent asymptotic expansions cannot exist within the same region, and the 
regions are separated by the lines or areas of zeros or poles of the function. The 
theorem is true whether the function be a quotient of repeated or non-repeared 
integral functions, with zeros of simple or multiple sequence. 
We need only consider the case of integral functions—the general theorem will 
follow, since every function of the type just 
mentioned can be represented as a quotient 
of two integral functions of finite genre. 
The zeros of the function must proceed 
according to fixed laws, and therefore, in our 
diagram of the region near infinity, they will 
mass themselves infinitely close together as 
we approach infinity itself. They will 
therefore form certain lines (not necessarily 
straight) or areas of ultimate singularity. 
If the areas entirely surround 2= 00 there 
will be no asymptotic expansion possible. 
We thus assume that there exists an area 
such as co AB, non-shaded in the figure, 
within which, if the radius 00 A is sufficiently small, there are no zeros of f (z). 
Suppose first, that the zeros of the function form a single simple sequence, and are 
non-repeated ; then it may be written 
y 
F (2) = e H(2) n (l — - 
n=i . a n 
P - 1 /_tV" 
& n= 1 m \««/ 
e H(s) <f> (2), (say), 
where p is the genre ’ (independent of n), and H (2) is a holomorphic function. 
Suppose that 2 lies between circles of radii ja„| and j«„ +1 | where n is very large, 
then those terms of the product <f> (2) for which \z \ < a w+1 may be written, as in the 
proof of Weierstrass’ fundamental theorem, 
e r,C) 
where P 3 (2) is a function represented by a series of positive powers of 2. For those 
VOL. CXOIX.—A. 3 M 
