206 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
§ 20. We have obtained the asymptotic expansion of Gp(x; 0) in the two cases 
when lv (x) > 0 and when iv (sc) < 0 by separate methods. By this, however, we are 
left in doubt as to the behaviour of the function on the imaginary axis. We proceed 
now to obtain the two expansions simultaneously. 
We shall limit ourselves to the case kl($) > 0 , as the result can then be extended 
to all values of 6, except those which are zero or a negative integer. 
As previously, we have 
G s (sc; 0) = tr ^ ~ f (— zf" 1 exp{— z0 + e~ 2 x}dz, 
277 J L 
where now, since iX(0) > 0, the contour L can be taken to embrace the positive half 
of the real axis. 
Hence 
Gp (sc ; 6) = d [ (-^y 3-1 exp {—z6+e~ 2 x}dz + —^ f s' 3-1 exp { -z0+e~ z x} clz. 
2it J i \P) ^ v 
The first integral is taken round a circle of radius rj surrounding the origin, 
beginning and ending on the positive half of the real axis, which is a cross-cut to 
render (—s )^ _1 uniform. The second integral is taken along the real axis. 
If iLv (fi) > 0 , the first integral vanishes as 77 approaches zero. For simplicity we 
consider only this case. The limitation simplifies the statement of the proof : its 
absence does not affect the argument. 
By the substitution 1 —yjx — e~ z , we now obtain 
the integral being taken along the straight line from O, the origin, to B, the point x. 
