and provided that in Equations [^Bajb] ojt is changed to tot + ex in all A 

 terms but to wt + B in all B terms, a and 6 being independently arbitrary. 

 Equations [Ul+a,b] require no change. 



(c) Calculation of a' , a" , etc. The coefficients a' , etc., in 

 11 11 11 



Equations [^3a,b] and [UUa,b] can be calc\ilated by evaluating the 



coefficients of A and B in Equations [U2a,b] and separating real and 



imaginary parts. In preparation for this calculation, each complex number 



may be replaced by the sum of a real and an imaginary part. Write 



q = q' + iq" ; q = q' + iq" 

 ^1 ^1 ^1 2 ^2 ^2 



in which q' , etc., represent qr or qB/(2r) in Formulas [35a, d]; then 



a' - iq" q' - i<l" 



1 1 ^1 ^1 12 2 



- " q' + iq" ,2 ^ „m2 ' - 2,2 



q^ ^1 ^1 q^ + q2 q2 q' + q;; 



and from Formulas [3Tm-p] applied to the definitions of s , c, S, C: 



s = sin q' I cosh q" 2. + i cos q' I sinh q" Jl 



c = cos q' I cosh ql' )!- - i sin q' Z sinh q" I 



S = sinh q ' Jl cos q" X- + i cosh q ' Ji sin q " X. 



C = cosh q'l cos q" J, + i sinh q ' 2. sin q"l 



Similarly, Q , Q , sind Ql can be separated into parts by using Equations 

 [33a] and [3l+a]. 



Denominators can be hsindled most conveniently by transferring i to the 

 numerator. Write D = D' + iD" , determining D' and D" from Equation [hi]. 

 Then: 



1+1 



