90 Proceedings of the Koyal Society of Edinburgh. [Sess. 
which is valid, when E(0)>O, for the same path as the integral in (2), we 
find that 
where 
1 ! 2 ! ^2 
{m^-{k-\f} . . . {m^-{k-s + m 
+ • • • + (.-!)! .-1 
^ _ 1 Ii{k-^ + m) 
* r( J — A: + ??i) s ! U{k — ^ + m — s)z^ 
From this formula it can be deduced* that ^s^(p{z)lz^, where (j>{z} 
remains finite for all values of 0 such that i/a — tt < amp 0 < \//- + tt, 0=^0. 
When R(m — /<: + J):j>0 we can obtain this result by using the contour 
integral expression for ^(0), as in Whittaker and Watson’s Analysis^ 
16'3, and making the contour approach infinity in the direction 
amp^ = 'i/A. Thus the asymptotic expansion of W^, ^(0) is valid for 
— 37 t/ 2< amp0<37r/2, 0^0. 
Wo,„(20), it follows that the asymptotic expansion 
of K^(0) is valid in the same region, while the asymptotic expansions of 
G^(0) and J,^(0) hold for — tt< amp0<27r, 0^0, and —ir< amp0<7r, 0^0, 
respectively. 
Since K„(0) = J 
PART II. 
A Fourier-Bessel Expansion. 
In a previous paper *j* the author has given a proof, depending on 
contour integration, of the validity of the ordinary Fourier-Bessel expan- 
sions. A similar proof will now be given of the validity of the expansion f 
where 
/(r) = 2,A.RW, 
s 
^{k) = J n(Kr)Gn(Ka) - G„(k?-) J n(xa) , 
and Ks is a positive zero of 
S(^) = J ^(Kb)Gn(Ka) - Gn{Kb)J n(Ka) . 
To begin with, let us assume that the expansion is valid for a<r<b; 
then, if 
Q(^) — tJ fi(^KX^G , 
* Prof. G. A. Gibson, loc. cit. 
t Proc. Edin. Math. Soc., vol. xxxix. 
1 Cf. Gray and Mathews’ Bessel Functions^ chap. x. 
