250 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OP 
Part VI. Page 
The function fp ( x ; 0 ) = 2 /*+ ’ when 1*1 < 1 . 282 
} i= o \n + v)p 
Part YII. 
The function 2 ^ (1 + --- ) - (0<«<1). 283 
The function S .283 
n = 0 r(l+»l) 
The function S 0. 284 
„=i 1(1 +n + nO)’ 
Possibility of further extensions... 28 ^ 
Part VIII. 
00 X 71 OQK 
The function &(*) - 2. r(1 : ' + —“ > . .- 
00 X n OQQ 
The function E.(*;«,« “,1 0 r( i+»*)(«.■ 
Part IX. 
Notation of generalised hypergeometric integral functions... 292 
The function iFj {a, p; x} = ^ 2 T ^ ) ■—t and the properties of the allied function 
1 u 1 r(a) n= o r (n +1) r (p + ?i) 
« 1_p iFi{a^/) + l;2-p;a:}. 292 
Part X. 
The function 0 Fi {p ; x} = T (p) 2 —- f! -- . The asymptotic expansion of J„ (x) for 
Lr> J 71=0 r («+1) i \p+ n ) 
complex values of n and . . 29 ^ 
Part XI. 
General theorems on hypergeometric integral functions. 29 ^ 
Introduction. 
§ 1. Integral functions can be defined either by Taylor’s series or Weierstrassian products. V hen the 
zeros are simple functions of their order number, the latter method is, as a rule, most simple. V hen the 
zeros, however, are transcendental functions of the order number, those integral functions which so fai 
have occurred in analysis have been defined by Taylor’s series. 
[Definitions by definite integrals have usually been reducible to one of the preceding forms.] 
Whatever be the manner of its definition, an integral function has a single essential singularity at 
infinity, and the behaviour near this singularity serves to classify the function. By studying this 
behaviour we may hope to find connecting links between the two modes of definition. 
The behaviour at infinity is determined by asymptotic expansions. 
The first expansion of a function was derived from Stirling’s 1 approximation to n !. This led naturally 
to expressions for P (%) when x is large and real. 
1 Stirling, ‘Methodus Differentialis,’ 1730. 
