288 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
Now, if q = 2p +1 , we have 
sin Tnjm/a _ 1 — 
sin 7 rm/a 1 — e _2irt ” l/a 
p 
_ S' g2jufi»i/a 
P=~P 
Hence the integral is equal to 
E a (*)-(l/a) | exp {* 1 /a e Wa }. 
Now, if s = 'it+i/y, the integral will tend exponentially to zero when |u| is large if 
—la 7 T+ |2p+l — a| 7 T — 7 T+ |arg*| <0 . ...... (1). 
We suppose in the integral that * s = exp {s log a:}, and we take that value ol arg a; 
which is such that | arg * | < tt. 
If now a > 2 jp -f 1 , we obtain from the condition ( 1 ) 
fair -2 (_p+l) 7 T+ |arg*| < 0 , 
and this will hold for all values of | arg * | which are < tt, provided f-a < 2 p+ 1 . 
On the other hand, if a< 2 jp+l, we must have 
2 p 7 r— §ua+ | arg * | < 0 , 
and this will hold for all values of | arg * | which are < tt if 2p +1 <f a. 
The contour integral will therefore vanish, when taken around that part of the 
circle at infinity bounded by s = —Jc, when k is a finite positive quantity, for all 
values of | arg * [ which are < tt if we take p such that either 
\cl < 2p + 1 <a, 
or 
a < 2p + 1 < fa. 
In either case therefore the integral will be equal to 
4 r (am) ( — ) m sill TT (a —q) m ] T 
m =1 
77* 
where, when \x\ is large, | J*| is ol lower order than 1/|* 
AVe therefore have 
p \ 
E a (*) - t - exp 
a 
} = 
A 
S' 
1=1 r (1 — aw) * 
- + J* 
• • (C). 
This asymptotic equality is valid for all values of arg* between ±77 (the limits 
included) if p be so chosen that either 
“< 2 p+l<«, 
Li 
or 
a <2 p+ 1 < fa. 
