the quotient is equal to the nth asymptotic coefficient, designate it Cj^. Note that the C's 

 are numbered from to N. Let the coefficient of y^ in Bj^ be Bjj ^. Then 



N 



'N+j 



%,N-iCN+j-i 



1 <j<N 



(82) 



i=l 



Here the C's are numerical constants. The unknowns, the a's and b's, are in the terms of B. 

 Suppose that these unknowns have been determined up to n = N-1. Then eq (82) will 

 contain two unknowns, ajsj and bjsj. By using eq (82) for j = N- 1 and N, the unknowns can 

 be evaluated. The index N can then be increased by 1 and the process repeated. The 

 process can start with N = 2 if aj, bg, and bj are provided, but these are easily determined 

 from eq (81). The terms of B^ are determined from eq (80), which gives for each term 



^n,m " ^n-l,m-l +bn^n-l,m + ^n^n-2,m • 

 Any Bn m i^ ^^^o i^ ^n is greater than n. 



(83) 



When j - N-1 is used in eq (82) in the process described above, the coefficient of 

 the (2N-1) power of x is being evaluated. This term is expected to contain a^ and b]\j, but 

 — as will be proven later — because the coefficient of bjsj is zero, ajsj is the only unknown in 

 a linear equation and can be easily evaluated. The next term determined with j = N contains 

 ajv[ and b^, but now only h-^ is unknown and is easily evaluated. 



As an example, the Cj^'s through n = 10 are listed in table 1 . These are the asymp- 

 totic series coefficients given by eq (72). The corresponding ajj's and b^'s as determined 

 above are also listed. A more complete Hst of the a's and b's can be obtained from the 

 FORTRAN program in appendix C. 



Table 1. 



Asymptotic series coe 

 continued fraction 



fficients, Cj^, and the corresponding 

 coefficients, a^, and bj^. 



n 



Cn 



^n 



bn 







1. 







1. 



1 



0.10416 



0.10416 



-0.80208 



2 



0.08355 



-0.58764 



-2.28555 



3 



0.12823 



-2.29072 



-3.77864 



4 



0.29185 



-5.11525 



-5.27462 



5 



0.88163 



-9.06285 



-6.77193 



6 



3.32141 







7 



14.99576 







8 



78.92301 







9 



474.45154 







10 



3207.49009 







30 



