MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
447 
On pursuing the same course as before we find that the given series is the 
asymptotic expansion of the function 
y + log ( *) + j y 1 dz. 
But there is the important difference that now the integral which has led to this 
result has no pole along the line of integration. And log (— x), instead of being 
allowed to take any one of an infinite number of values, has such a value that 
log (— a?) is real when x is real and negative, and has for complex values of x whose 
7T 
real part is negative an amplitude which lies between + — • 
We see then that the process employed has led, when 2ft (x) is positive to an 
infinite number of functions, all of which have the same asymptotic expansion ; and, 
when 2ft (x) is negative, to but one such function. 
Evidently when we seek an asymptotic expansion for the function* 
ao yj "+1 
x) = e~ x % -- 
' r=1 r . r ! 
we may say that we get, when 2ft (x) is positive, 
f( x ) 
1 ! n ! 
1 + —+... + —+• 
for terms like x{y + log (— x) + 2rmn}e * are negligible compared with the least 
term of the asymptotic series ; but when 2ft (x) is negative, we get 
1 ! n \ 
f(x) = e~ x x{ log (- x) - y] + 1 + — 
in which successive terms are of decreasing order of magnitude. 
The zeros of the function f (x) near the essential singularity x = oo , are ultimately 
along the imaginary axis. 
We thus have an illustration of two important propositions :— 
(l.) A uniform integral function may admit of asymptotic expansions of different 
form in different areas with their vertices at its essential singularity. 
(2.) These portions of the plane are separated by lines of zeros of the function. 
§ 43. Inasmuch as in Parts III. and IV. of this paper we proceed to actually 
obtain asymptotic expansions satisfying these laws for all the most simple types of 
* I was asked to investigate this function by Mr. G. W. Walker, Fellow of Trinity College, who 
desired to compute it in certain physical researches. Originally I obtained the expansion by considering 
the differential equation x 2 + y = x, in a way bearing great resemblance to that employed by Horn, 
‘ Crelle,’ vol. 120, pp. 17 and 18. 
