PERTURBATION THEORY FOR EXPONENTIAL M CURVE 189 
of them to a p-th one. The absolute values are then 
determined from the normalization condition 
/ U,%(2) dz =1= Aa (7) 
Let therefore 
Crx =1 5) (55) 
and the p equations in equation (24) are just suffi- 
cient to determine the (p — 1) constants C,, and 
D,. We divide the equations in (28) by A, and pick 
the k-th equation (n = k) to solve for D,, while the 
other equations are used to solve for the Cym, as is 
illustrated in the scheme below for the particular 
case of k = 1. 
D, & D,° 
= = — — Bu — Cr Bie — C13 Bis — re (56) 
a a 
(2 - 2 + pu) Op 
a a 
= — Bir — Cis Bos — Cis Bu — - ~~, (57) 
(2 = P+ te) Cr 
a a 
= — Bis — Cie Bos — Cis Bar — Tene a) (58) 
(2 - 28 + pu) Cu 
a a 
= — Bu — Cie Ba — Cis Ba — AA een aS (59) 
As a first approximation one puts 
0 
Ds — By = Bu ? (60) 
a a 
erase Bie 
0 
(2: age pss) GD 
a a 
Cy = — Mesa SB etc. , (62) 
0 
(2 - 28 + Bus) 
a a 
where the value of D,/a obtained from equation (60) 
is used in equations (61) and (62). Next, one substi- 
tutes these values of the C’s in the right-hand sides 
of equations (56) to (59) and resolves for D:/a and 
the C’s. This procedure has been found to be rapidly 
convergent and is, furthermore, self-correcting in 
case of arithmetical errors. 
EXPANSION OF D,; INTO A 
POWER SERIES IN a 
When a is small, it is convenient to expand D, 
into the series 
D, = DD. +aD, +0?D,@ ++++. (63) 
It is known from standard perturbation theory that 
2 
DD = — Bex; D, = e'*/8 > eee i) eee : 
* - (T™! —? Tk) 
mek. (64) 
It is possible also to derive an alternative expression 
for D&?: 
3 
1 ONS 6 
DEN ag 
= (4 
24/d 
il = D,° =z (23/12) 
e . 
0 
[( i x) Dy (X +2) + De (A) Dy @ |ve a 
Dy (X) = 5 Ds ()? + 
1 poe d+ (93/12) | 
Ay/)\ - 
i [ > Q) + (2 a *) Di (+ »\va dz. (65) 
=D, (A)? + 
Since the former expression is simpler for compu- 
tational purposes, we shall not give here the deriva- 
tion of equation (65). 
APPLICABILITY OF PERTURBATION 
METHOD TO A MORE GENERAL CLASS 
OF M ANOMALIES 
It is possible to apply the results obtained for the 
case when the M anomaly is of the form f(z) = ae-* 
to more general types of M anomalies. To begin with, 
if 
fle) = ae + ye, (68) 
then we merely write in equation (28) in place of 
OBnm(A), [@Bnm(A) + Bnm(u)]- Once the Bnm(X) are 
computed as functions of i, there is no additional 
labor required to deal with an f(z) which consists of 
a sum of any number of exponential terms. If instead 
of f(z) = ae-* we had {@ = aze-, then the corre- 
sponding 6’,,(A) would be 
Bran (0) = | Unt) Unde) 20% de = — Ban) (67) 
0 
Tf Bam(A) is known, dBam(A)/ddA can be computed 
directly from equation (43). When equation (43) is 
integrated numerically, the derivative dBpm(A)/dA is 
computed at each point in any case. Evidently, for 
T(z) = az*e-“, where k is a positive integer. 
Blam (A) = il U,,%(z) U,%(z) 2* e™ dz 
es (—) a*Bnm (A) ? 
dy? 8) 
