502 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1954 



In order to obtain expressions corresponding to (13.13) for Un(z), T^n(^) 

 we examine Fig. 11.6(d). We have already seen that the limits of in- 

 tegration for s, in the integral (13.11) for Vn(i^'^p), are 

 [co exp (t27r/3), oo] . In the same way it follows that the limits for 

 Unii^'^p) and Wn(i^'^p) are [=» exp (— t27r/3), cc exp (z27r/3)] and 

 [ CO , CO exp (— t27r/3)] , respectively. When we take s' = s exp (+ i2Tr/3) 

 as new variables of integration (with the upper sign for Un(z) and the 

 lower one for Wn (z)), the integrals corresponding to (13.11) go into 

 Airy integrals. 



We can write our results for z = i^'^p, when n is close to ip/2, as 

 follows : 



Vn{i"p) ~ Ci~"'Ai{U^"), (13.14) 



Wn{l"p) ~ Ci'"Ai{hl-"'), 



where 



c = (p/4)^^^(27r)''V'''^^Vr(^^^) , 



h = i2/pr\m - ipV2)z-''\ 

 i = exp {iTr/2), m = n -\- 1. 



(13.15) 



The asymptotic expansions whose leading terms are given b}^ (13.14) 

 may be obtained by the method used by F. W. J. Olver"^ to study 

 Bessel functions. 



Ai{;x) and its derivative have been tabulated for positive and negative 

 values of x* Here we shall use the definitions and results as set forth in 

 Reference 11. These tables and (13.14) enable us to obtain values of 

 Un{i^''p) along the rays in the //(-plane defined by arg {m — ip'/2) = tt/G 

 and — 57r/6. Along the tt/G ray hi"'^ is negative. Since the tables show 

 that the zeros of Ai{x) occur when x is negative, it follows that the 

 zeros of Un{i'~p) occur on the tt/G ray. In the same way it is seen that the 

 zeros of Vn{i''p) and Wn{i''p) occur on the 57r/6 and the — 7r/2 rays, 

 respectively. This agrees with Fig. 12.1. 



The Airy integral defined by 



* Reference 11, page 424. 



28 Some New Asymptotic Expansions for Bessel Functions of Large Orders, 

 Proc. Cambridge Phil. Soc, 48, pp. 414-427, 1952. 



