274 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
We proceed to show that the function 
Y ( x . m — £ x Y ( n + @) 
a to(n+eyr( n +i) 
may be written as the sum of functions 
oo 
X b r G p+r {x ; 0), 
r =0 
and to obtain its asymptotic expansion when | arg a: | < tt/ 2. 
We have at once 
F, {x ; 0) = £ — t ( n + e y+r 1 
A 
■s' 
+ X 
J k + r 
If then k > R, we have 
CO 7 
O Wt + r 
r-1 (n + 0y + * + ’ 
» jf/A + r 
< |(n + 0y + *| |(n + 6>) r 
< 
V k 
V r 
(n+ey +k | r“i n' 
where y is the minimum value of |n + 0|. 
The modulus of the series is therefore less than 
V k 
+i 
— , since y > V. 
Hence 
where 
| (?i + oy +k | (p- —0 
F„(a;; 0) = X 6 r G„ +r (re ; 0)+J*, 
p+i 
r=0 
|J,|< S 
a? 
»=o n! |(» + ^r A |(/x-0‘ 
Hence | J A | tends to zero as k tends to infinity, since \n + 0\>V. We therefore 
have 
F„(»; 6) = 2 6 r G (+r (®; «). 
)•=0 
the series being absolutely convergent for all finite values of | as |. 
§ 31. We will next show that , iw/i-ew | arg x | <7t/2, 
F p (a;; 6 ) — 
S, 
.+ 
J.W 
a^ [ s =o r (1 — — s) a; 5 af J’ 
w/iere S s = X & w *c,-»r(l-/8-m) and \ J a (x) | tends to zero as |j»| tends to infinity. 
m =0 
The coefficients r c n are given by the expansion , valid when \y\ <1, 
log (i —y )~ 13+r ~ 1 
-y 
(1 -yT x = t r c n {- y y. 
