496 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
separated by zeros of its derivative. [It cannot, however, be proved that the derived 
function has not other real or a (necessarily even) number of other imaginary zeros.] 
This theorem I shall not prove, as it is not connected with the main developments of 
the present paper. We proceed, however, to show that such developments complete 
and to some extent verify this extension of Polle’s theorem, and that incidentally 
they furnish many criteria as to the nature of the derivative of a given integral 
function. 
93. Let us consider, as an elementary example, the function of genre zero and 
order — , 
P 
P„ (*) = n 
',i=l 
1 + 
IV 
, where p > 1. 
We have the asymptotic equality 
P p (z) = (2 t t) 2 z 1 exp. 
7r 
sin 
7 r 
ZP d~ 
( -y-'Fipsi 
Pemember, now, that it has been proved in Part II. that we may differentiate an 
asymptotic equality of this type, and we obtain 
f r p p( 2 ) = ( 27 0 22 -e x P 
^ zl + . . . 
7 T 
Sill - 
P 
7T 
I=£ 1 
Z p — -p . . . 
7r pv ' 
Sill - 
t P 
J 
x_P 7T l±p | 7T 
= ( 27r ) 2 u ^ 2 p ex P 
ZP 
+ 
together with terms whose ratio to the terms 
sm - 
sm 
P L p 
retained tends to zero as | z | tends to infinity. 
From this expansion we see that 
(1) P p ' (z) is of the same order as P p ( 2 ), 
(2) The zeros of P/ (z) are such that, when n is large, we have with the usua 
notation a n = n p + (p — l)?i p-1 + lower terms. 
Not only so, but theoretically, by finding successive terms in the expansion for 
P p ' ( 2 ), we ought to be able to determine the form of its n ih zero as nearly as we 
please. Practical difficulties will, of course, arise when we come, in the asymptotic 
expansion, to a term which arises from a transcendental term in the n th zero of P p ' ( 2 ). 
Note that the formula for a u may be readily verified when p — z. 
For 
and, therefore, P 3 ' ( 2 ) = 
n 
n=\ 
sinh 7 T\/z 
7 T \/Z 
jL cosh 7 r^/z x sinh 7r z 
T*S Z 
— ^ _., /;2 asymptotically. 
z 
