DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 349 



Now substitute in this result the value of z Aj /_2, ? +i ; i" the result thus 

 obtained the value of z iti _ s>g+ i , etc. After a certain number of substitu- 

 tions we have : 



]■ 



c k, I c k, l+l— I 



The lower limits c k> t will be determined later to satisfy several con- 

 ditions. 



We may choose at pleasure any one of the m elementary divisors, say 

 (r — r K ) e *, and then select any one of the corresponding e K solutions of 

 (18) for the first approximation. We shall take then for the first ap- 

 proximation z^/,0, the values given in (11) for a particular A; the inte- 

 gers k and A will remain fixed for the solution we are now developing. 



For the development of the solutions corresponding to (r — r K ) 6K , we 

 shall make the following further assumption as to the coefficients b' k> t . 

 Let us examine all the exponents of the elementary divisors (r — r A ) c *, 

 which are such that Er A = Br K , where Rr k means ''real part of r k " 

 and pick out one exponent, say e K , that is as great as any one in this 

 special set of exponents ; i.e., 



(23) e K >e k , where Rr k = Rr K . 



The assumption is that the integrals : 



/itfnu-r^* ()zl:!:::::) 







converge. If in particular r K is a simple root, and no multiple root has 

 the same real part as it, then e K — 1 = 0, and this further restriction 

 drops out. Or it might happen that r K is a multiple root, but that all the 



