MR. E. W. BARNES ON INTEGRAE FUNCTIONS. 
415 
That is to say, for all points in the region of 2 — qo which are not at a finite 
distance from the zeros of n~ l z~ l sinh tt ^/z, this function admits what we may call 
the asymptotic expansion ('2n)~ ] z _i and a similar property is true of all functions 
of the first class. 
§ 3. Consider next the second class of functions. 
We have as the simplest example 
1 
r (z) 
e y ~ z n 
1 
1 + — e 
\*-± r 
Since the time of Stirling it has been known that, when z is a large positive 
integer, 
T (z) = (2?r)*. 
'=* e 
-■■+ 2 Bs + 1 J 
s=0 2s + i . 2s+2 
approximately—the terms neglected involving exponentials of a lower order than 
those retained. 
In 1889 Stieltjes* proved that tins asymptotic expansion is valid for all values 
of 2 in the region of z — oo , excejit those which are at a finite distance from the 
zeros of T _1 ( 2 ). 
By a different method it is possible to establish both Stieltjes’ result and the 
analogous theorem that the double gamma function! 
ur 1 ( 2 ) = e i] ~^ +y2 \ z . n ri 
OO GO 
1 + a 
o n + u n- 
n A = 0 ni — 0 
t 
in which n = m l b} ] + m. 2 o ) 2 , admits the asymptotic expansion 
log 
F, (z) *ST(o) 
p 2 (&)j, o> 2 ) 
8 S' A ( 2 ) { log Ui+w 2 — 2 (to + to') 7 TL 
I 2 s < 2 d0i -L - s f 1 +11+ N ( >"V s *+d°) 
+ • 2 ^ U + 2) -h m {m + 1)zm ■ 
This expansion was shown to be valid for all points in the plane of the complex 
variable 2 near infinity, which are not at a finite distance from the zeros of the 
integral function Tp 1 ( 2 ). 
A similar theorem is true for multiple gamma functions. 
§ 4. As regards the elliptic functions and the integral functions associated with 
them which constitute the third type, there are no points in the neighbourhood of 
infinity which are not at a finite distance from the zeros of the function and no 
asymptotic approximations are known to exist. 
* ‘ Liouville ’ (4), vol. 5, pp. 425-444. 
t See a paper by the Author, ‘ Phil. Trans.,’ A, vol. 196, pp. 265-387. 
