500 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
j ( z ) . J n ( z ) 
we see that all the zeros of - ±L rr L are real if the zeros of ——- are real—a theorem 
/yll-T 1 
<v A/ 
due to Macdonald. # 
If the function P (2) be multiplied by an algebraic polynomial with real coefficients 
whose degree is q, the derivative of the product can at most have only p-\-q imaginary 
roots. 
§ 98 . So far we have only considered integral functions whose roots are all real and 
negative. If, however, we have an integral function all of whose roots lie along 
a line other than the negative half of the real axis, a change of the independent 
variable will at once reduce it to an integral function all of whose roots are real and 
negative. 
If then an integral function of genre p have all its roots but q lying in a sequence 
along a straight line through the origin, its derivative will at most have p + q roots 
which do not lie along this line. 
§ 99 . We now conclude for the present the applications of the expansions which 
have been obtained. There are many questions which are still to be discussed—for 
instance :— 
(1) Functions of infinite order ; 
(2) Functions of multiple sequence ; 
( 3 ) Asymptotic expansions deducible from linear differential equations ; 
(4) The rate of increase of the coefficients of the Taylor’s series expansion of an 
integral function ; and so on. 
Investigations in connection with each of these questions have been tentatively 
undertaken—notably by Borel, Horn, Hadamard and Poincare. And I find it 
possible to extend, by the methods of this memoir, many of the results which have 
hitherto been obtained. But such investigations I leave for future publication. 
* Macdonald, “Zeros of the Bessel Functions,” ‘ Proc. Bond. Math. Soc.,’ vol. 29, p. 575. 
