498 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



Table 12.3 — Leading Terms in the Asymptotic Expansions for 



Un(z), Vn(z), W^(Z) WHEN Z = l^"'p, p > 



regions in the m = n + 1 plane corresponding to the different asymptotic 

 expressions are shown in Fig. 12.2. The boundaries are simply those of 

 Fig. 11.2 reflected in the real /M-axis. The lines of zeros are also shown in 

 Fig. 12.2, and are reflections of those of Fig. 12.1 except for the inter- 

 change of Vn{z) and Wn(z). 



Table 12.3 may also be constructed by returning to the paths of in- 

 tegration shown in Section 11. It may be shown that corresponding to 

 every path of steepest descent for z = i^'^p, n = rii there is another 

 path, obtained from the first by reflection in the real /-axis, which gives 



the path of steepest descent iov z = i p, n = Wi 



The notation used in Table 12.3 is as follows: 



z = i~^''^p, m = n + 1, i = exp (iV/2), 



-37r/2 ^ arg m < x/2, -3x/4 ^ arg to < t/4, 



-37r/2 ^ arg (-zp' - 2m) < 7r/2, -57r/4 ^ arg ti < 37r/4, 



t, = [r^'> + i-ip' - 2m)"-]/2, k = [i'~p - i-ip' - 2m) ^'1/2, 

 Ao = [k"\-ip - 2mT"'/{-2iir"')\ expfito), (12.9) 



A[ = [h"\-ip' - 2m)-^'V2x '1 expM), 



f{to) = zto -\- ^ - m log /„ = -( 1 - log - - log - 



m 



zh + 



m 



m 



2 "2 



Sometimes it is helpful to use 



m log /i = -I 1 - log - - log - 1 + i ph . 





-1/2 



{2^r 



exp 



2 



log 



7n 



1/r (j^^-^j = l/r(l + n/2) for -x/2 < arg m < t/2, (12.10) 

 27r = r+'r(-n/2)/2x, -37r/2 < arg m < - 7r/2. 



1 — m 



