MR. E. AY. KARNES ON INTEGRAL FUNCTIONS. 
425 
the first h — 1 terms for which | z | > <£ (n) being omitted. Thus it converges with 
If 2 "j m x or 
Exp. 
ill m 1<£0)I 
The modulus of the term inside the bracket is less than 
OO 00 
S? V 
m = 1 n = /.• 
1 n ) I _ 
mx f n) 
$ 00 
x(») 
1 - 
2 
X (n) 
1 4> 0) 
Now 1 — 
</>(”) 
xOO 
(n — -f- 1 
< 2 
n=k 
co ) 1ms for its greatest value a finite positive 
2 X («> 
converges, which is 
quantity A (say). The product then converges if 1 - 
n=h I 9 \ n ) 
ensured by the condition assigned at the outset. 
The function whose existence has thus been established has y (n) zeros on a circle 
of radius j (f> (n) |. If, since the assigned condition makes the order of c f> (n) greater 
than that of y ( n ), the zeros will ultimately be separated by arcs of infinite length. 
§ 22. A little ingenuity will enable us to construct other functions of types 
innumerable, among them what Borel has called functions “ cl croisscmce irre- 
guliere The survey gradually forces upon us the conclusion that we cannot 
expect to find any general law as to the behaviour of all integral functions near their 
essential singularity which is not a disguised truism, t MM. 11 a dam aril; and Borel 
have given laws relating to the increase of all integral functions. It seems to me that 
such laws rfiust be limited to particular classes of functions, and that such delimitation 
cannot be stated too explicitly. Consequently in this memoir I have taken the most 
simple functions and have endeavoured to study in detail their behaviour near the 
essential singularity, for I believe that by such means the progress made will be sure, 
if slow. 
* Borel, [‘Fonctions Entieres,’ Note III.], gives an example of such a function in the form of a 
Taylor’s series. 
t Such a term I should apply to M. Borer’s law “the maximum value of a function is equal to the 
inverse of its minimum value on an infinite number of circles at infinity.” For this law is an immediate 
consequence of the possibility of asymptotic expansions (see Part II. of this memoir). 
| Osgood (‘ Bulletin of the American Math. Soc.,’ Nov., 1898, note, p. 80) states that the analysis used 
to prove IIadamard’s most general law requires revision. And it is to be noted that Hadamard 
(‘ Liouville,’ 4 ser., t. 9, p. 173) assumes that </> (m) is continuous, increasing, and such that L<£ (m) + ^ 
constantly increases ultimately. 
O 
o 
VOL. CXCIX. 
A. 
