478 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
It therefore possesses an asymptotic expansion which is the limit, when e 
vanishes, of 
(2") exp j sing(p _ e) 
TTZ?-* 
+ UAp * 
V 
x /—y—i j s 
N' ^ ’ F 
+ 
p-ej s= - p+ i sz° 
P—e 
= ( 27T ) 
z 
'* ex P \ (- Z Y l °g z + ~ 11 + ^ f v ^ 
p 
S=z—p +1 
= (2 IT) -is-* exp {(- *)' log * + (-r-’f + F 
i(*) + r - 
= y + terms which vanish 
remembering that F (— 1) = y, and that L t 
S=1 
when 5=1. 
We thus obtain the same asymptotic expansion as in the previous paragraph. 
Note that we have obtained our expansion by making p increase up to the nearest 
integer. If, on the contrary, we make p decrease down to the nearest integer, there 
is no breach of continuity in the introduction of an additional exponential factor. 
Thus we have 
(z) = Lt n (l -f- 
e = 0 n — 1 
—— + .. . + 
1 
,.P+ e 
pn 
V 
p+e 
n 
P+e / 
and therefore we have the asymptotic expansion 
_ 7 TZ P + e 
['Zn) z * exp \ -t— 
c = 0 
(z) = Li (27r) 2 pif 2 i exp 
+ 2 
/ (-) 
t-i 
sin 7r (p + e) 1 S= L P szf \p + e, 
Now, unless s — — 1, F (s) = £ (— s) ; and therefore we have asymptotically 
t) i x T . /„ f(— z ) p (l + elog* + . ..) (—z)p ( p 
B, ( 2 ) = L« (2tt) 2r+ . * exp |---+ — £ [— 
+ S' - 'ph F f--X- 
= (Shr)-*.- exp {(-:)- log « + f(£ 
the same expansion as before. 
This paragraph is instructive in that it shows how the asymptotic expansion calls 
for another exponential factor in each term of Weierstrass’ product as the order 
passes through an integral value. 
§ 72. If now it is desired to construct a function which is the natural extension 
among simple integral functions of the ordinary gamma function, we take 
r (zip) 
= e 5=1 * 
s F(~i) A I /. . 2 \ nr+ • < • + 
I 1+ 5® le ” 
<-*f 
Up 
pH 
p/p 
