4(54 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
But, because of the validity of the Maclaurin sum formula, it includes simple 
functions with very rapid convergence—such as those for which 
€ n c*i ' 
a„ — e? X function of n of lower order than e e ■ . 
§ 58. We can now extend the result which was obtained in § 52, and find an 
asymptotic approximation for a simple integral function with an algebraic sequence 
of zeros, that is to say, of a function of which the rd h zero, — a n , admits, when n is 
large, an expansion of the form a n — n p -f- b- l n p ~ e1 b. 2 n p ~ eo - + . . ., where p is greater 
than unity and the quantities e l5 e 2 , . . . are real positive and in ascending order of 
magnitude. 
We take a n — (f> (n) - r so that, by reversion of series, 
I i Lm p + i _ 2e a h 3 h ^ , , / x , N 
n — vp - -vp +- - - - r p - -tp -f . . . = i If (r) (sav). 
p p z 2 p \ / \ »/ / 
Since, when s is positive, we may expand a n s directly by the binomial theorem, 
we have, when m is large, 
ni— 1 
Hi — 1 
2 a n = 2 
,i=i ,i=i 
n ps -j- 61 + ^ n ps 2e ' -j- sb z n p5 e - . 
f m 
4>' ( n) cin + F ( ps) -j- sb l F ( ps — e 2 ) + 
AO) , | 
2 ^ £ 0 
W 1 V F (ps-tej+sb, F (ps-*,) + ... 
( -)' B,+, d‘>» 
2t + 2 ! drn? t+1 
<t>' (™) 
}> («) dn + F (ps) + Z (p, s ; £ l ■■■)- * + y ( ; 
i-y b, +1 d 2t+i 
2t + 2 ! dm~ t+l 
<f) s (m), 
where Z ip, s ; 
e 2 • • • 
O &2 • ■ • 
is a definite finite quantity vanishing with the quantities b, 
And 
>i=i 
! . h h 2 , h . 
p log n -j- — yy -(- ■ -(- . . 
which can be expressed in terms of a series of Riemann £ functions. 
Again, when s is positive, 
v __L_ — |‘ m is/ \ i WO ) s (—yB <+1 d 2<+1 
n = m <f>s(ri)~ \ $ W W 2 E 2f+~2l dv^t 
m— 1 m —1 
2 log cf) (n) = 2 
=i >i=i 
= ["log 4, (n) dn + A log 2ir + 6, F (— E ,) — y F ( — 2e,) + 6 S F (— % ) 
log O) , £ (-)*B /+1 d 3m 
- 2 +,? 0 (yT2i! lo s * W > 
and wo shall put Z (o ; "'[ = F (-«,)- A F (_ 2£] ) + /,. F (- t,) + . . 
