DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 349 



Now substitute in this result the value of 2^, /_o ^,+1 ; in the result thus 

 obtained the value of z^j_3^^^i , etc. After a certain number of substitu- 

 tions we have : 



X 



r,. 



(21) z,,,,^, = x'-^^j^-dx...Jx-' 



^i. L. g+1 "-^ 



'^i. L+1 



t=l—L ^ 



+ 2 j -^^ • • • j ^"'■^■2 2 C.i-,^„..^x , 



where L <i I. 



When Z = we have : 





X X 



t=l , , ^ i=m j—e- 



(22) 



*^, /, g+l 



== ^''^' 2 / -^^ • • • / ^"''^ 2 2 *^//+w2:,,y,^rfx. 



'-■/t, / ^k, l+l—l 



The lower limits c^, ; will be determined later to satisfy several con- 

 ditions. 



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

 (r — O*''' ^"^ ^^^"^ select any one of the corresponding e^ solutions of 

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

 proximation ^x, /, o> the values given in (11) for a particular A; the inte- 

 gers K and X will remain fixed for the solution we are now developing. 



For the development of the solutions corresponding to (r — O*"' ^^ 

 shall make the following further assumption as to the coefficients 5^'/^. 

 Let us examine all the exponents of the elementary divisors (r — ^^)% 

 which are such that Rvj^^^ Rr^, where Rr^. means ''real part of r^," 

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

 special set of exponents; i.e., 



(23) ^iT = ^ki where Rr^^ = Rr^. 



The assumption is that the integrals : 







converge. If in particular /•« is a simple root, and no multiple root has 

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

 drops out. Or it might hapi)eu that r^ is a multiple root, but that all the 



