188 TECHNICAL SURVEY 
PROPERTIES OF Bnm() 
For small ) the solution of the differential equation 
(43) can be started with a power series in }. 
1. nim 
nt 2heHn!s iowk/3 Xk (45) 
Bam(A) aah ne (t™ eae Tn)? fond Cre , 
6 
CO=1-,n= CaS ap? 
C= 10C, = 2 Gm =F Tn) 
tad Ga ra Tn)? : 
_ 14C2 — 2C) (1m + 1) 
Ys (Ga = a) ; 
Cn fe (4n =F 2) Cran = 2 (Gps Tn) Cro r= Cr : (46) 
(Ga i Tp) 
2n=m 
Bom = 14+ BiA4+ BorX?+---:, (47) 
2 0 = es 2(0) 
By = 3 Dn , B, 15? ; 
-~1, 8 pa) 
fn Te. 3 
pe ono p, BE || CS) 
n (Qn ae 1) m n— GQ) 
For intermediate values of \ one may either use 
the integral in equation (44) or integrate numerically 
the differential equation (43). The latter procedure 
was advocated by Hartree. 
For large values of \ an asymptotic expansion can 
be obtained directly from equation (43) by writing 
it in the form 
Bam (A) os 
2 ABum (X) 
2 de Oe 
ne + 2d (D2 + D2) — 2 + (D2 — Dd)? 
2 
N+ 2d (DB + D3) — 2 + 5 (DS — DB)? 
8 | a TL, (DD E52) = < (D8 is yy] 
+ 
[» + 2d (D2 + D8) — 24 i (Dn — Dy). 
(49) 
An alternative asymptotic expansion can be derived 
from equation (44) by partial integration 
Bmn— 
2 
[» + 2d (Dp + Dt) +5 (Dn — 3) | 
4 [a+ 20a + D9 - 309 — Dye] 
ae ate LE Pg Lp (50) 
[a+ 20 (D8 + Dy +4 (D8 — DY] 
In doing so one needs to prove that 
o 
3 
Gi FDR, + DA) = ig + a5 (Dh — DH)” 
0 Vx 
=V,, = 0. (51) 
We shall state here without proof that 
C 3 
dx — rp0_7_ 
Wee —— m 12 
l Va- 
TV 1 2\' 
= "YE (Z) hu (Da!) he (Dnt), (52) 
where h,; and he are Furry’s functions of the first and 
second kind defined as 
vee Ge) 
2) VETO G 2). (54) 
Since by definition of D,,.°, ho(Dn°) = 0, it follows 
that Yinm = 0. The proof of equation (51) for n + m 
is left as an exercise to the interested reader. 
hy (x) 
he (x) 
ITERATION METHOD OF SOLVING 
FOR THE CHARACTERISTIC VALUES D, 
AND THE COEFFICIENTS 4,, 
In solving equations (28) and (80), which are of 
infinite order, one proceeds by first assuming that 
Aim = Oform > p, where p is a convenient integer, 
and then evaluating D, and Ajm, m = 1,2--: p 
Next, one assumes that A,, = 0 for m > p + 1, 
resolves for D, and the A,,,, and the accuracy of the 
results is judged by the agreement between the 
values in successive approximations. The direct solu- 
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 p equations in equation (28), 
being homogeneous, do not determine the absolute 
values of all the A, but merely the ratios of (p — 1) 
