MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
485 
We note that the constants which enter will be expressible in terms of the 
coefficients of p„ and values of the Riemann £ function. 
We might now investigate the asymptotic expansion of a repeated function of 
finite order less than unity with algebraic sequence of zeros of the type 
a» = n p 
A &.•> 
'+h+?+ 
where the quantities e ls e 3 , . . . are real, positive, and in ascending order of magnitude. 
The analysis is, however, such an obvious extension of the corresponding result of 
Part III. that it may be at once supplied by the reader. 
Repeated Simple Functions of Finite Non-integral Order greater than Unity. 
§79. We next consider the asymptotic expansion of the function 
F (*) = n 
n=l 
, , z Y" % l/z v> 
1 + —) e n m hj m (" a) 
'd'n / 
where p < p < p + 1, and p is such that 
2 is convergent, and 2 divergent, 
CC n 
/J'n 
when e is a small real positive quantity. 
The analysis is an obvious modification of that employed in § 66. 
We find F (z) = F 0 z M e /(f)+ ,*L p N , where a n = <f> (n), y (m) = | p„ dn, 
in-1 rm 
2 dn + M — p m + . . . , 
51 = 1 j 
5JI—1 Cm 
2 Pa log (f) (n) = P„ log <f> (n) dn — log F 0 — \ y m log <f> (m) + . . . , 
n = 1 J 
771—1 
71 = 1 
5 - X Uly 
2 p„f s (n)= p„dn + F,— -f . . . (s=—p, — (p— 1),. . . — 1, 1, 2,... oo ), 
and 
f(z) = Lt i f x [i \j (-zef] 
k= x> J 
sin ( 7 j + j) 
sin I <j> 
Icf). 
§ 80. As an example, we may consider the function 
CT+ 1 
F (z) = n 
71=1 
, where is not integral, and p < __ <_p +1, 
The order of the function is —-—-. 
