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MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
functions. Here for real values of the variable Stokes 1 first gave asymptotic expansions, and Orr 2 has 
recently extended his results to general complex values of the argument. 
Quite recently Mittag-Leffler 3 has constructed the new function E a (x) and investigated its 
asymptotic behaviour. 
It was, however, in the theory of linear differential equations that POINCARE 4 first pointed out the use 
of divergent series as solutions in the neighbourhood of infinity, and laid the foundation of a rigorous 
theory of such series. The continuation of his investigations has been the subject of many researches, 
notably by Kneser and Horn. For references in this connection I may refer the reader to Forsyth’s 
‘ Theory of Differential Equations.’ 5 
Another connected series of investigations may be mentioned. Hadamard 13 first gave a remarkable 
theorem as to the maximum value of the modulus of an integral function defined by Taylor’s series on a 
circle of large radius. Other theorems of similar type are due to Borel 7 and BoUTROUX. 8 Valuable, 
however, as such theorems are on account of their generality, we need complete asymptotic expansions 
before we can adequately classify integral functions. Further references will be found in Borel’s 9 
text-book. 
There is a close connection between the asymptotic expansions of certain types of integral functions 
and what Borer 10 has called the associated functions defined by Taylor’s series of finite radius of 
convergence. This connection enables us to investigate the singularities of many types of such Taylor’s 
series, and thus connects the theory with a whole series of investigations. Reference may be made to the 
work of Fabry, 11 Le Roy, 12 Lindelof, 13 and Leau. 14 A very complete bibliography of this branch of 
modern mathematics is given by Hadamard. 15 
§ 2. In the present paper the author attempts to give unity to the investigations of asymptotic 
expansions of integral functions defined by Taylor’s series by taking various standard types of such 
functions and applying new methods of contour integration so as to get, as simply and elegantly as 
possible, complete asymptotic expansions. For each function investigated we find the nature of the 
behaviour at infinity. The investigation may be regarded as preliminary to the formation of a classified 
table: it is complementary to that previously carried out for functions defined as products. 
It is hardly necessary, perhaps, to say that no methods, however powerful, will apply to every function 
that can be constructed by a Taylor’s series. Just as, in general, a Taylor’s series admits its circle of 
convergence as a line of essential singularity, so the general integral function, which we may define by a 
Taylor’s series, will not admit the same dominant asymptotic expansion for any range of A r alues of arg x , 
however small. 
1 Stokes, ‘Cambridge Phil. Soc. Proc.,’ vol. 6, pp. 362-366. 
2 Orr, ‘Cambridge Phil. Soc. Trans.,’ vol. 17, pp. 171-200; pp. 283-290. 
3 Mittag-Leffler, ‘Comptes Rendus,’ vol. 137, pp. 554-558; ‘Acta Mathematica,’ vol. 29. 
4 Poincare, ‘Acta Mathematica,’ tome 8, pp. 295-344. 
5 Forsyth, be, cit., Part III., vol. 4, 1902, p. 341. 
6 Hadamard, ‘Liouville,’ ser. 4, vol. 9, pp. 171-215. 
7 Borel, ‘Acta Mathematica,’ vol. 20, pp. 357-396. 
s Boutroux, ‘Acta Mathematica,’ vol. 28, pp. 1-128. 
9 Borel, ‘Le§ons sur les Fonctions Entieres,’ 1900. 
10 In the memoir just cited. 
11 Fabry, ‘Annales de l’Dcole Normale Superieure,’ ser. 3, tome 13, pp. 367-399; ‘Acta Mathematica,’ 
tome 22, pp. 65-87 ; ‘Liouville,’ ser. 5, tome 4, pp. 317-358. 
12 Le Roy, ‘ Annales de la Faculte des Sciences de Toulouse,’ ser. 2, tome 2 (1900). 
13 Lindelof, ‘Acta Societatis Scientiarum Fennicae,’ tome 24, No. 7. 
14 Leah, ‘ Liouville,’ ser. 5, tome 5, pp. 365-425. 
15 Hadamard, “La Serie de Taylor et son prolongement analytique” (‘Scientia,’ 1901), 
