420 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
The standard reduced simple integral function with a single simple sequence of 
non-repeated zeros is thus 
n 
51 = 1 
p » _1 i . , V. 
_ T \ «m= i m v ' 
We shall call this briefly a simple integral function. 
^12. The quantity p„ is the smallest integer such that the series N a~ p ‘ is 
71=1 
absolutely convergent. When the convergency of the series can be assured by 
taking for p d some number p independent of n, the function is said to be of finite 
gem e # p. In this case, if p is a real positive quantity such that S 
71=1 
p + e 
converges and 
71 = 1 
— e diverges, however small the real positive quantity e be, 
the function is said to be of order t p, and p is called the convergence-exponent j of the 
series — , — It is sufficient that the function ct n depends uniquely 
Cl i Cl o Cl ii 
upon n ; if we put a n = (f> (n), the quantity <£ in) is not necessarily a uniform function : 
it may be a definite value of some multiform function of n. 
CO 1 
§ 13. When there is no finite quantity p which will make the series N-r; 
converge, the function is said to be of infinite genre and infinite order. The con¬ 
vergency of the series can, as Weierstrass first showed, always be obtained by 
taking p = n. A theorem due to Cauchy proves this at once, since 
If 
A 
/if 7. = 
0. 
It is equally sufficient to take p = log n, for then S , —r l0 ^ = 
71 = 1 
— V 
1 
7i=i n 
log | a, | 
; and the 
latter series is convergent, since | a„ | increases indefinitely with n. 
But a smaller number still is a sufficient value for p, namely, the greatest integer 
contained in —, where e is any positive quantity as small as we please.§ 
The great difficulty in the theory of asymptotic approximations for functions of 
infinite order consists in finding the minimum value of p. I do not intend to consider 
such functions in the present memoir. Functions of the type cf I{:) , where Id ( 2 ) is 
holomorphic, are of course integral functions of infinite order. 
§ 14. It is evident that if a n does not increase more quickly than some (possibly 
fractional) power of n, however small, the associated integral function will be of 
* Laguerre, ‘ Comptes Renclus/ vol. 91; ‘ CEuvres,’ vol. 1, pp. 167 et scq. 
t Borel, ‘Acta Mathematica,’ vol. 20, p. 360. 
X vox Schaper, ‘ Hadamard’sclien Func-tionen,’ p. 35 ; Borel, 1 Fonctions Entieres,’ p. 18. 
•§ Borel, ‘ Acta Mathematica,’ vol. 20, p. 360. 
