432 
MR. E. AY. BARNES ON INTEGRAL FUNCTIONS. 
The point 2 = 0 will be an essential singularity of the function of which the 
integral is the formal expression. For certain values of 2 near 2 = 0 the integral can 
probably take values which differ infinitely for the smallest change in the value of 2 . 
This will happen when 2 lies on a line of zeros or poles crowding to 2 = 0. Along 
such lines, or quite possibly within areas of the same nature, the asymptotic series 
will cease to represent the function. 
Further, there must always be such lines or areas of non-representation, for the 
only functions to the essential singularities of which poles or zeros or other 
^ 1 
1 
singularities do not crowd are of the types e p , c e .... , which cannot admit of 
asymptotic expansion. 
We have then the fundamental result that the integral cannot represent the 
“ sum of the series right round 2 = 0. There will be certain lines or areas with 
2 = 0 as extremities or vertices along which the asymptotic series cannot be “ summed 
by any process which we may employ : these lines or areas will differ with the 
different processes, but will never be absent altogether. 
There are, of course, asymptotic series of the prescribed type which can never be 
ao 
“ summed ” by any process which we may employ. Such a one is N a c z Cn , in which 
n = 0 
L t (c„ +1 — c,) = =c and L t ^'ci n = 00 . But such series will never arise naturally in 
n = co v. = 00 
analysis, and we do not, therefore, need to trouble about them. 
§ 30. We have now to consider whether, when a series of the prescribed type is 
“ summed ” by means of the process indicated, the function which results admits the 
series as an arithmetically asymptotic expansion according to Poincare’s definition. 
Denote by f{z ) the integral which is the result of summing the series 
a o + a \ z + • • • + a ,i~ l -f- • • , 
_ n / 
in which L t \/a n = 00 and I J - v c “ = 0. 
n = x ' t \ = :o % 
The associated function for the series is 
Cf ( 2 ) — a 0 c 0 + « 1 c 1 2 + . . . + a„c n z’ 1 + . . , 
and for s„, the sum of the first n terms of the series, is 
«o c o + «i c F + • • • + 0^*3*. 
w 
Hence 
hi ere 
f(z) - _ 1 r G„ +1 (z, ■?) 
z“ 2 sin 7T0 J z " 
c 1 (— x) e " J dec, 
-"G„ +1 ( 2 , x) = [a,fi n -f a„ +l c, l+l (xz) + ...]. 
