284 
ME. E. W. BAENES ON THE ASYMPTOTIC EXPANSION OP 
III. When | arg x\ > ^ (1 — a) 7r, we may 
2 
Case I. 
IV. The expansions of Cases I. and III. may 
contour integral 
o 
obtain the same expansion as in 
also he obtained by considering the 
— fr (— s) (— cc)T (a.s+ 1) ds. 
lm J 
Y. When a = 1/2, we obtain the asymptotic expansion 
fix) = Fe^xV (i) + £ 
lAy n = 0 
( —) n r( 2 >i + 2) 
F(n+l)x 2n 
valid for all values of arg x, if P = 0 when |arg x \ > tt/4. 
VI. If 6 (x) denote the function Z d>0; we have 
r w »=i T (1+n+nO) 
<t> (x) = f xy e exp [xif (1 -y)} dy. 
J 0 
Hence, if 1\ (.x)> 0, 
d r 
<f>(x) = exp [xO fl /{6 + 1) 0+1 } \/2 {6 {B+1)/2 x V2 /[6 +1) {9+2)/2 } x |E (|-) t 
the d,’ s being definite constants. 
YII. We can deduce this result from the result of I. by means of the contour 
, If T (— s) x7T ds 
integral — -— —-A—A- j- -t— . 
& 2m J r ( — as) sm tt (l — a) s 
( — as) sin tt (1 — a) 
YIII. By considering the contour integral —Y I —/ —'-ffx • 
J *=> ° 2 771 j r (1 + s + Os) S 111 TTS 
that, when | arg x | < 7t/2, 
1 f x T(l + 0sl _E_ dSi we call show 
j , , . S aTT (n+nff) 1 ; r(l+ r t +(w + l)/0 
<H-x) r( to j-- eZ r(i +») 
n =1 
Thus, when H (:»)<0, <j){ x ) needs two asymptotic series for its expression. 
IX. The previous result can also be obtained by combining the results of III. 
and VII. 
X. Similar analysis can be applied to series of the type 
* T(l-n + q0n) , n 
i r ( 1 + On) 1 
where 0> 0, and q is an integer. 
