A simple computer program given in appendix C will find |z| for each value of k from 1 up 

 to the maximum number of terms desired. The largest term, m, is easily determined because 

 from one k to the next m will remain the same or increase by 1 , so it is only necessary at 

 each step to check term m + 1 to see if it is larger than term m. 



The FORTRAN subroutine HANKEL given in appendix A uses the above power 

 series method to compute hj and h2 for small arguments. The coefficients a, b, c, and d are 

 given in lists by that name. The truncation points are given in the list called ZMLA2, which 

 lists values of |zK determined by eq (59) or the three similar equations. 



ASYMPTOTIC SERIES EXPANSION USING CONTINUED FRACTIONS 



When the argument z falls outside the curve in figure 4, hj and h2 can be computed 

 more efficiently or more accurately by asymptotic series than by power series methods. 

 Reference 1 1 gives information on branch cuts and regions of validity of the two forms of 

 the asymptotic solution (Stokes' phenomenon). Here we will give computing formulas that 

 comply with these requirements, without discussing them further. 



Since a given expansion is valid in one or more quadrants, we choose complete quad- 

 rants as regions. For z in quadrants 1 , 3, or 4 use 



h2(z)~exp(57ri/12)F2(z) (60) 



h2(z)~exp(-7ri/12)G2(z) (61) 



For z in quadrant 2 use 



h2(z)~exp(57ri/12)F2(z) + exp(ll7ri/12)Fj (z) (62) 



h2(z)~exp(-7ri/12)G2(z) + exp(-77ri/12)Gi (z) (63) 



For z in quadrants 1 , 2, or 4 use 



hj (z)~exp(-57ri/12)Fi (z) (64) 



h'l (z)~exp(7ri/12)Gi (z) (65) 



For z in quadrant 3 use 



hj (z)~exp(-57ri/12)Fi (z) + exp (-IItt i/12) F2 (z) (66) 



h'l (z) ~ exp (tt i/12) G] (z) + exp (In i/12) G2 (z) (67) 



The four auxihary functions follow: 



M 

 Fj (z) = K z-1/4 exp (2i z^l^/3) S Cj^X^" 



m=0 



(68) 



25 



