MR, E. W. BARNES ON INTEGRAL FUNCTIONS. 
4(13 
fvn * ( _y-i ry« 
— i l°g 2 + j log (p (n) dn — S —-—j ft (n) drt 
+r + f ft o+M^+ 
1 7 (—)*<* , (-yg* 
2 s 1 
Since <£(»i) is a very large quantity we may utilise the expansion of if* (t). 
integral last written is, therefore, equal to 
The 
[4> («‘) 
2 a, 
)•=0 j ./■ 
* (w) clt J 1 , i /(-)V S (-y* 
1 ' ii \ Z? ^ t s 
t p 
— N 
i 
0 <£' p (;/t) r 
, 4/ (-)' . ^ 
1 "r “ , r' 1 
- s — r + - 
P 
— 
a. 
!' 
7T 
?" r 
7T - — >• 
= 2 (-)'tc- 
■'■=° ! sin — 
L p 
> ( 
i 
sin — L (— z)p J 
P 
IT 
IT 
ZP 
" f y° 
If, then, we introduce the Maclaurin constants, log F 0 = log <£ (r^) cZn, 
— F, = | <f) s (n) dn, we shall obtain the asymptotic expansion 
n 
;s=i 
i + 
z 
= F 0 z * exp < 
ird 
is (— z ) 
</>(»_ 
. 7r 
sin - 
. ( - *) 1/p . 
L P 
| ( — F" 1 F, 
s = l SZ* 
Such values of the many-valued functions introduced are to be taken as would be 
indicated by the analysis. 
§ 57. It is evident that the investigation of § 54 applies to all simple integral 
functions whose primary factors need no exponential to ensure convergency. Thus 
it includes all simple functions of order -, where p is real positive and > 1 with 
algebraic zeros. It includes all simple functions with noil-algebraic zeros of the type 
given by a lt = [ an p -f bn Pl + . . .] (l T n) a , where r and p are both real positive and 
> 1, cr is positive or negative; where l T (n) denotes log {log { . . . n] }. . . }, these 
being r repetitions of the logarithm, and where p, p l5 . . are decreasing quantities 
tending to — cc as a limit. 
