418 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
His principal proposition is, ‘ If, as 2 tends to co 
found for which the limit of 
<£' ( 2 ) 
<M 2 ) 
a very great value of 1 2 1 can be 
tends uniformly to the value zero, then cf) ( 2 ) is of genre n! 
Shortly afterwards, Poincare # gave further criteria for the genre of a function, and 
made the important step of pointing out that the near connection between the genre 
of the function and its behaviour near infinity lead to an approximate determination 
of the magnitude of the general term of the Taylor’s series for the function. 
After a succession of notes by Cesaro,! Vivanti| (who proved that the derivative 
of a function is of the same genre as the function itself), and Hermite,§ the subject 
remained in abeyance until Hadamard,!] in a memoir crowned by the French 
Academy, gave a valuable extension of Poincare’s results. 
The latter had proved that in the Taylor’s series for an integral function of 
genre E, the coefficient of x m multiplied by the (E + l) th root of m ! tends to zero, as in 
indefinitely increases. 
Hadamard proved that, if the coefficient of x m is less than ( —V , the function is, in 
\m \] 
general, of genre less than X. He also showed that when the coefficient of x m is of 
order , where X is not an integer, the function represented by the series is of 
genre E, designating by (E + 1) the integer immediately superior to X. 
Finally, Borel,11 continuing Hadamard’s researches, introduced a more precise 
notion than that of genre (f § 12), and attacked the difficult problem of functions of 
infinite order whose convergence is very slow. 
[Note added March 2 Oth, 1902.] In his recent text-book, “ Le 9 ons sur les 
Fonctions Entieres,” ## Borel has given a valuable precis of our present knowledge of 
integral functions. And a paper by MELLiNff has recently come to my notice, which 
should be carefully read by all interested in the subjects with which the present 
memoir deals. 
§ 10. The present contribution to this interesting theory differs from previous 
investigations in that it is shown to be possible to substitute actual asymptotic 
equalities for the inequalities which have been previously obtained.]]; 
* Poincare, ‘Bull, des Sciences Math.,’vol. 15, pp. 136-144. 
t Cesaro, ‘ Compt. Rend.,’ vol. 99, pp. 26, 27. 
| Vivanti, ‘ Battaglini,’ vol. 22, pp. 243-261, and 378-380; vol. 23, pp. 96-122; vol. 26, pp. 303-314. 
§ Hermite, ‘ Battaglini,’vol. 22, pp. 191-200. 
|| Hadamard, ‘Liouville’ (4), vol. 9, pp. 171-215. 
U Borel, ‘ Acta Mathematical vol. 20, pp. 357-396. 
** Paris, Gauthier-Villars, 1900. 
ft Mellin, ‘Acta Societatis Fennicae,’ 1900, vol. 29, No. 4 
[ft Mellin has obtained results of this nature.] 
