296 
MR, E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
+ (q — 1) other terms similar to the last, admits the asymptotic expansion 
r (« t ) r (a l -p l + 1)... r (cx,- Pri +i ) ^_ a , 
r(l-a 2 + a 1 ) ... r(l—a / ,+ a 1 ) 
x 
i +1 
a 1 , «i-pi + 1,..., «1 -pq+ 1 ; a! —a 2 + 1,..., «i —a p + 1 
II 
xj 
provided | arg x | < (q—p + 3) tt. 
There are evidently p relations similar to the one just written, each corresponding 
to an asymptotic solution of the differential equation (1) of p in the neighbourhood of 
x — oo. There are therefore q+l—p other asymptotic solutions near x = cc which 
will be asymptotic expansions of other linear combinations of the q +1 fundamental 
hypergeometric functions. 
[II.] The linear combination of hypergeometric functions 
n r(i-p r ) 
^- pFjloq ,..., CL P ) p u ...,p g ; (-) p+l+q x} 
n r(i-a r ) 
r —1 
, r(p r -i)n'r(p r -p 1 ) 
+ S X 1 Pl ---—-?F ? {1— oq — p,-,..., 1 +a p — p r ; 
’ = 1 UT(p-a t ) 
2-p„..., pg-pr+l ; (-) ?+1_p ai} 
can be expressed by the contour integral 
r(-s) n r(i-/o r -s) 
- ——- x s ds, 
IT r (1 — a r — s) 
and provided j arg x \ < (q+l—p) Trf2, this integral may be taken along a line in the 
finite part of the plane parallel to the imaginary axis. 
If q+l—p = p, we obtain 
where 
when 
exp {px l/ *} (2n) a ^p} 2 1 = — -J— ( S (s) x s ds, 
2ttl J 
S (s) = t 
r (s+t/p.) n T(i-p r +t/p-s) 
)■ = ! 
t = 0 ^ -r, [t + r 
V- 
n r 
p 
t 
n r ( l —ot } , t — — s) 
= 1 \ [X 1 r= 1 /x 
+«i+... + « ? -p 1 -...-p ? |//x. 
