456 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Unfortunately we may not say that the sum of an infinite series of integrals is 
equal to the integral whose subject of integration is the function to which the series 
of subjects of integration can be “summed.” The two integrals last written down 
can then only be evaluated by reducing them to an exceeding complicated extension 
of the type known as Dirichlet’s integrals. The analysis is utterly intractable. 
If we make m 2 = ze~ w , and expand the subjects of integration in powers of 2 , then 
we can say that the last two terms will not contribute terms whose order of 
magnitude when 2 is large is comparable with that of any positive or negative power 
of 2 . And, as we know, the sum of these two terms is equal to log (1 — c -2 ^). 
The formula log F ( 2 ) = tt?} — log (2irz*) is thus asymptotic exactly as the Maclaurin 
in— 1 
series for m ! and X n 2s , from which it is derived, are asymptotic. 
n= 1 
That is, for large 
values of [ 2 1, the expression z~ n {logF ( 2 ) — irtfi + log27r2 i ] for all values of n tends 
to zero as 1 2 | tends to infinity. There is, in fact, Poincare’s arithmetic asymptotic 
dependence. 
The preceding example will serve to show the nature of the asymptotic expansions 
which we can now proceed to obtain. 
§ 52. We consider first the function P p ( 2 ) = n 
n=\ 
, where p > 
1. 
We have 
log P p (:) = (»i - 1) log 2 - p 
m — 1 
log n 
ni — 1 
+ 
>1 = 1 
00 
. . . + 
. . . + 
(-r 
-1 ps 
n 
sr 
(-rv 
sn ps 
t • • .. - 00 1 /)l ^ 
Therefore, if we substitute the approximations for log m — 1 !, X ~ p ,, and X n ps given 
n=m H ?i = l 
by the Maclaurin sum formula, we shall obtain the expansion, arithmetically 
asymptotic with regard to m, 
log P p ( 2 ) = (m - 1) log 2 - p 
| C-r 1 j m pS+1 _ 
s= 1 *3'' 1 p s + 1 
_j_ ^ (~rv J m ~ ps+1 
4=1 s [ps — 1 
(m - |) log m - m + log ^2^ + X - ' -w 
y — ] —/ . —j / _L 11L 
-ps 00 
. , V ' B '-_ (~P S 
2 r =i ps + 2r — 1 \ 2 r 
m - P s- 2.- + 1 
