SOLVING VALUES /'<, AND THE COEFFICIENTS A Um 



253 



An alternative asymptotic, expansion can be derived 

 from equation (44) by partial integration 



/3 ran — » o 



\» + 2A(Z>° +£>£)+ 5; (Z>° - D° n y- 



+ 



5X a + 2X (£)Ji + D°) - ^ (Z>° - I>°) 2 



In doing so one needs to prove that 

 d.r 



E 



*„m = . 





We shall state here without proof that 



dx 



\Lr = / ip 



* mm I /— & 



h y/x 



_ 



m 12 



iVt /2 



Ai (A. )'** (Z>„°) 



(50) 



tion of equations (30) and (28) is, however, a labo- 

 rious process which rapidly increases in complexity 

 as p exceeds about 4. The following iterative pro- 

 cedure has been found effective and of the same 

 intrinsic simplicity for any value of p. 



To begin with, the /) equations in equation (28), 

 being homogeneous, do not determine the absolute 

 values of all the A km but merely the ratios of (p — 1 ) 

 of them to a p-th one. The absolute values are then 

 determined from the normalization condition 



/ m{z)dz = 1 = V\a, 



Let therefore 



(51) 



ft -"-km p i 



W-m — a > ^ kk *■ 7 



■n- kk 



(7) 



(55) 



(52) 



and the p equations in equation (24) are just suffi- 

 cient to determine the (p — 1) constants C km and 

 D k . We divide the equations in (28) by A,. k and pick 

 the fc-th equation (n = k) to solve for D kl while the 

 other equations are used to solve for the C km , as is 

 illustrated in the scheme below for the particular 

 case of k = 1. 



where hi and 7i 2 are Furry's functions of the first and 

 second kind defined as 



= Pll — Cl2 P\i — Cj.3 /3i3 



a a 



(56) 



Ai (x) = 



hi (x) 



Vx~Hi 



Vx Hi 



(i) it 



(53) 



(54) 



Since by definition of D m a , hi(D m °) = 0, it follows 

 that fy mm = 0. The proof of equation (51) for n ± m 

 is left as an exercise to the interested reader. 



256 ITERATION METHOD OF SOLVING 

 FOR THE CHARACTERISTIC VALUES D k 

 AND THE COEFFICIENTS A km 



In solving equations (28) and (30), which are of 

 infinite order, one proceeds by first assuming that 

 A km = for m > p, where p is a convenient integer, 

 and then evaluating D k and A km , in = 1, 2 • - p. 

 Next, one assumes that A km = for m > p + 1, 

 resolves for D k and the A km , and the accuracy of the 

 results is judged by the agreement between the 

 values in successive approximations. The direct solu- 



£>i Z>2 



a a 



Di D, 



a a 



+ Pa C» 



Pu- c 



13 P23 — ^ 14 P24 



Ci 



, (57) 



— C12 Pis — Ci4 Pa — " ' * 7 (58) 



+ fe C 



a ■at 



= — Pu — Cu Pii — C-13 Pu 



As a first approximation one puts 



Dl - Dl ° B 



— Pll 7 



a a 



C12 — — 



Dl £>2° 



a. a 



+ Pi 



• (59) 



(60) 



(61) 



