M 



F2 (z) = k z-1/4 exp (-2i z^l^l3) S C^ Y"^ (69) 



m=0 



M 

 Gj (z) = kzl/4exp(2iz3/2/3) V Dj„ X"^ (70) 



m=0 



M 



G2(z) = kz^/4gxp(_2i 2^/2/3) S T)^Y^ (71) 



m=0 



where X and Y equal + i z ' ^ respectively, and 



K = 21/^31/67^-1/2 = 0.853 667 218838951 

 The coefficients Cj^ and Dj^^ are again computed by recursion relations where Cq == Dq == 1 : 



Cm = Cm-l [9(2m-l)2-4]/48m (72) 



and 



Dn, = Dj„_i [9(2m-l)2-16]/48m . (73) 



Square roots of z are to be taken so that the real part of the root is always positive and the 

 imaginary part has the same sign as the imaginary part of z. This applies also to fourth 

 roots. The three-halves power is obtained as the product of z and its square root. 



The summations in eq (68) -(71) can be done as indicated or evaluated by contin- 

 ued fractions. When done as indicated they are asymptotic series, and care must be taken to 

 truncate them at the term of smallest magnitude, if this term is reached, because adding 

 more terms will reduce the accuracy. Since the largest term in these series will always be 1 , 

 the series can be truncated if the terms become less than \Q~^ in magnitude, where h is the 

 number of decimal digits in the computer word. 



Continued Fraction Expansion 



The method of continued fractions is more effective in evaluating these asymptotic 

 series, and it is used in subroutine Hankel in the FORTRAN program in this report. The 

 coefficients are stored in lists entitled C4, C5, D4, and D5. In the remainder of this section 

 the continued fraction technique is presented, along with the method of determining 

 coefficients. 



The continued fraction has the form 



F(x) = bg + aj 



X + b 1 + a2 . (74) 



x + bo + . . . 



26 



