EVALUATION OF Prmi(X) AS AN INDEFINITE INTEGRAL 



251 



while for large z one may use asymptotic expansion 

 of equation (19) 



//i ( '-'(«) -^x/- 2 < " 



I- Si 12) 



1 + 



03 



385 

 L0.368m 2 



(26) 



wc shall study the function 



F{z) = U n °(z) UJ(z) , (31) 



where 



CV(*0 + [2 + DJ] UJ(z) = , (32) 



t//(2) + [^ + £>„"] U n °(z) = . (33) 



By multiplying equation (32) by U n °(z), equation 

 (33) by UJ(z) and subtracting, we obtain 



If now the expansion (16) be substituted into 

 equation (15), we obtain, on making use of equation 

 (17), the condition 



dz 



(UJ U n ° - UJ U n °) = - (DJ - D n <>) UJ U n ° , 



(34) 



°° r- 



(D k - DJ) + ae 



UJ(z) = . (27) 



On multiplying this equation by U n °(z), where n is 

 any integer, and integrating from to <» we get a 

 system of equations for the determination of D k and 

 the A , m : 



£ 



2 A km (D k - DJ) 5„ m + a p nm (\) = , 



n = 1,2,3 • • • (28) 



»(X) = / UJ(z) U n \z) e- X2 dz . (29) 



77 TJ _ 77 77 



= - (Z) m ° - D n ») f UJ(x) U n °(x) dx . (35) 

 Now it can be verified by direct substitution that 

 "F + 2F (D n + D m + 22) + 2F 



= (DJ - ZV) (UJ U n ° - UJ U n °) 



= - (DJ - TV) 2 f F(x) dx . (36) e 



From equation (36) it follows that 

 d 



F = 



dz 



(2 2 + DJ + ZV) F + ±f\ 



The characteristic values Z) A . are then obtained as 

 the roots of the infinite determinant. 



D k - ZV + ap n , «/3» , aPu , 

 a/3 21 , Z>, ; - D 2 ° + a/3 22 , a/3 23 , 



1 



+ o(A, 



Z) K °) 2 F(z) dx 



(37) 



a/3si , ^ 32 , D t - Z) 3 ° + afe , 



We may also note that 



F(0) = 2UJ(Q) U n °(0) = -2, (38) 



. 



(30) 



e- Xz Fdz = e~ Xz (F + \F) 



+ X 2 / e" Xz ZAfe 







Having determined D l: from equation (30), the 

 A km are obtained by solving the system of linear 

 equations (28). 



(39) 



25.4 



EVALUATION OF fi nm (\) 

 AS AN INDEFINITE INTEGRAL 



The primary task in the perturbation method is 

 the evaluation of the exchange integrals p nm (X) 

 defined in equation (29). We shall accomplish this 

 by proving that /3„ m (X), as a function of X, satisfies 

 a differential equation of the first order for which 

 an explicit solution can be given. For this purpose 



e This is the first occasion in the author's experience where 

 use is made of the fact that the product of two functions, 

 each of which is a solution of a distinct second order ordinary 

 linear differential equation, satisfies a fourth order linear 

 differential equation. 450 



