The coefficients are derived as follows. The well-known recursive relations that give 

 the Nth stage of a continued fraction as a rational function are used (ref 13). 



FnCy) - AnCyVEnCy) , (78) 



where 



N 

 i=0 



^N 



Je^Y^ 



(y + bN) An-1 + ^N An_2 (79) 



N 



-1 f./ 



i=0 

 = (y + b^) Bn_i + a^ Bn_2 , (80) 



in which A_i = 1 , Aq == bg, B_] = 0, and Bq = 1 . Again y = 1/x. The long division indicated 

 in eq (78) is then carried out, giving a quotient in terms of aj, bj, and y that can be equated, 

 term by term, to the first 2N-1 terms of the asymptotic series. 



The long division is carried out with Ajvj and Bjvj written in descending powers of y. 

 The quotient is then in descending powers of y or ascending powers of x. Fortunately, the 

 first 2N+1 terms determined for any N are identical to the same initial terms for any larger 

 value of N. This will be proven later. The first few equations obtained from the division are 

 as follows: 



bo = Cq 



aj = Cj 



-ai bi = C2 



aj ( b, -ao 1 = C^ 



ai^aajbi-b^ + ajbj) = Q (81) 



From these equations aj and bj can be determined, since the coefficients Cj are 

 known. However, a simpler method is available. 



The long division indicated in eq (78) can be carried to 2N+ 1 valid places; but be- 

 yond N+1 places, terms from the original dividend are no longer entering the remainder. 

 Therefore terms in the later part of the quotient have a simphfied form. Since term n+1 of 



13. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ed by M 

 Abramowitz and lA Stegun; National Bureau of Standards Applied Mathematics Series, vol 55, p 19, 

 1964. 



29 



