Figure 4. Line in complex plane dividing the arguments for which the 

 modified Hankel functions are computed by (1) power series (inside) 

 and (2) asymptotic expansion evaluated by continued fractions 

 (outside). 



POWER SERIES EXPANSION 



In this expansion h] and h2 are given by two auxiliary functions f and g as 



hi(z) - g + i(3)-l/2(g-20 



h2(z) = g-i(3)-l/2(g_2f) . 



The auxiliary functions are given by the expressions 

 M 



2 ^^' 



m=0 

 M 



g - Bz ^ bj 



(48) 

 (49) 



(50) 

 (51) 



m=0 



where X= z^ ,A = 2^l^/[r(2/3)] and B = 2^/^/[32/^ r(4/3)]. The derivatives hj (z) and 

 h2 (z) can be derived by straightforward differentiation of eq (50) and (5 1) to give 



23 



