MR E. W. BARNES ON INTEGRAL FUNCTIONS. 
45 J 
(possibly non-uniform) function x(j (z) (say). The remaining (B) terms form the 
summable divergent series 
co A 
** z? * 
s=-p * 
[There will only be a finite number of positive powers of z since the genre of the 
function is finite. | 
We have then 
A, 
.. t(z) + 2 Tr - 
b (z) — e s=- P - 
In other words 
log F (z) — xJj (z) 
00 A v 
admits the asymptotic expansion 2 ~ valid for all points but those in the vicinity 
8= —p J 
of the zeros of F (z). 
§ 45 . The process just sketched will become much more clear when it is applied to 
various particular cases as in the following pages. The proof may, by mere verbal 
alterations, be extended so as to include functions of simple sequence with repeated 
zeros. 
A function with a finite number of simple sequences of zeros can be expressed as a 
product of functions, each with a single simple sequence. The logarithm of each of 
these functions will admit an asymptotic expansion, and the sum of such expansions 
will be the asymptotic expansion for the logarithm of the function. But terms of the 
category i fi (2) may be of different weight in different regions, separated by bands of 
/ 
zeros, and thus the asymptotic expansions may differ in such regions, as has previously 
been seen in the case of the integral function 
co ^•<•+1 
e~ x t -—- 
r= 1 r.rl 
§ 46 . The general theorem which has just been given may be proved pari passu for 
integral or meromorphic functions with multiple sequence. We refrain from formal 
proof, as the consideration of such functions is omitted from the subsequent develop¬ 
ment of this paper. 
Neither do I make any attempt to consider functions of infinite order, or expansions 
near isolated essential singularities of uniform functions. The difficulties which arise 
are all subordinate to the main necessity of limiting the type of function under con¬ 
sideration ; it seems doubtful whether it is possible to give any general theorem 
concerning integral functions and their behaviour near infinity, which will apply to 
every function which can be constructed. For exceptional classes must always be 
infinite in number compared with those which can be formally defined. 
3 M 2 
