MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
417 
The present memoir is devoted to the answer of this question; and the question is 
closely connected with other subjects of enquiry. 
§ 7. Soon after Weierstrass, in 1876, published his great theorem relating to the 
formation of uniform functions with assigned zeros, Laguerre remarked the funda¬ 
mental nature of the number of terms in the exponential function which is necessary 
to form the “ prime factor.” The number was by him termed the “ genre ” of the 
function ; and the questions at once arose :— 
“ Is the genre of a function equal to the genre of its derivative ?” 
“ Is the sum of two functions of the same or different genre a function of genre 
equal to the common genre or equal to the larger genre respectively ?” 
§ 8. Again, by Holle’s Theorem it is known that the real roots of any algebraic 
equation, <f> (x) = 0, separate, and are separated by those of <f>' (x) = 0. 
Is this true when f (.r) is an integral function ? 
Closely connected with this enquiry is the further one :—“ If the roots of <f> (x) = 0 
are all real, are those of f (x) — 0 real, in the case when (f) ( x ) is any integral 
function ?” 
Again, it is evident that the more quickly the zeros of an integral function increase, 
the more quickly will the Taylor’s series for the function converge. Can any con¬ 
nection be discovered between the magnitude of the coefficients of the Taylor’s 
series and the expression for the zeros of the function it represents ? In other words, 
if we are given the general term of the Taylor’s series for an integral function, can 
we approximately determine the nature of its zeros ? # 
All these questions fundamentally depend on the asymptotic approximation for the 
function. The nature of the latter serves to classify the nature of the integral 
function. 
History of the subject. 
§ 9. As already remarked, WeierstrassI founded the theory of transcendental 
integral functions by constructing functions with any assigned zeros. Laguerre^ 
invented the term “ genre ” to denote the number of terms in the exponential 
associated in the prime-factor—and for functions of genre 0 and 1 proved that the 
real roots of the transcendental integral function (j> (x) = 0 are separated by those of 
f (x) = 0 . 
He also proved, as Hermite§ had previously proved for = 7 — , that if the roots of 
<f> (x) — 0 are real, those of f (x) — 0 are real, provided <f> (x ) is of “ genre ” 0 or 1 . 
* This question is not formally considered in the present memoir, as the expansions which are obtained, 
although they will give closer inequalities than any hitherto published, must be still further developed 
before inequality can be replaced by that asymptotic equality which alone would be a complete solution 
of the problem. 
t Weierstrass, “ Zur Theorie der eindeutigen analyt. Funct.,” ‘ Gesamm. Werke,’ vol. 2. 
+ Laguerre, ‘ Compt. Rend.,’ vol. 94, pp. 1G0-1G3, 635-638; vol. 95, pp. 828-831 ; vol. 98, pp. 79-81. 
§ Hermite, ‘ Crelle,’ vol. 90, p. 336. 
YOL. CXCTX.-A. 
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