MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
44 5 
For in this case the bulb of the contour integral which expresses Z may be taken 
so small as not to include the poles of , and we shall therefore have 
Z = — % a r r ! 
-7T r=0 
dz 
Jr 
(- z)-r-i = V a r t(-v) = % 
a 
r=0 
( = 0 
2< + l 
( -r 1 Wl 
2t + 2 
This series is evidently convergent if (f> (z) is an integral function whose order is 
greater than or equal to 2, a condition which is equivalent to the convergency for all 
values of \z\ of the series 
t ci r V (1 + r) z r . 
It is evident that the Maclaurin sum formula will hold good in many cases in 
i I log, 
which (f> (z) is not a uniform function. If it be a function like z p or z 9 , or either of 
these functions multiplied by an integral function of order greater than unity, the 
Maclaurin formula will be valid if we suitably specify the branch of the function 
considered. Instead of attempting to tabulate such cases, it is perhaps better that 
we should go back to the genesis of the formulae when they actually arise. Applica¬ 
tions of the formulae which will be made subsequently in this memoir will usually be 
to cases in which <f> ( z ) has very simple values; and all general formulae will be 
tacitly supposed subordinate to what we may call the Maclaurin restrictions. 
§ 42. As a second example of the theory of asymptotic series we propose to try 
and find the function of which the series 
+ • 
+ — + . 
is the asymptotic expansion near the essential singularity x = co . 
We know that, if n be a positive integer, 
r (n) = f e~ ( t' l ~ l dt 
J 0 
where the line integral is taken along any straight line L from the origin to infinity 
which lies in the half of the z plane to the right hand side of the imaginary axis. 
Therefore the given expansion asymptotically represents the function 
CO 
where G (u) is the function which is represented by the series X u n , and the integral 
71 = 0 
is taken alone; the straight line L. 
O Cu 
