DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 367 



Now consider the following limit: 



(78) limit x~ r « (log x)- {e *~ x) y K .' A 



z=0 



= limit 2 2 ^^(log*)"^"^. 

 *=0 i— 1 Z— 1 



On the right, if we take the limit of each term in the summation, we 

 shall find that it is zero in every case except the one in which k — k and 

 / = e K (cf. (64) III.). We have, then : 



-i -xs 1 



(79) limit* r « (log*') {e « ) fi' K = A,K,e K(e _ X) r 



So we can write as a set of solutions of (1), corresponding to the ele- 

 mentary divisors (r — r K ) K : 



(80) y r = * r * (io g ^->,' A ^ A= : ; 2 ; e j, 



where the functions i/^ K ' A are continuous in the interval < x < c, and 

 such that : 



(8') ^' i Lo = -(^x)iA^- 



The constants A lkl are determined independently of the functions 

 a, .in (1) ; and therefore all that we have said on page 34G in regard 

 to certain sets of them as linearly independent solutions of the equations 

 (4) in the special case of a tJ — holds equally well here. 



In order to ohtain solutions corresponding to (r — r K ) €K * we assumed 

 in the treatment of the canonical system that \b'^\ ]log x\ K ~ was inte- 

 stable up to x — 0. Now, since the coefficients b' k ^ are linear functions 

 with constant coefficients of the coefficients a tJ , it will be sufficient, in 

 order to obtain solutions of (1) corresponding to (r — r K Y K , to assume 

 that \a,-j\ | log xf- 6 -"" 1 is intestable up to x = for all values of i andj. 



Our results may be stated as- follows : If (r — r K ) e " is an elementary 

 divisor q/"A(r), and if we consider all the elementary divisors (r — r k ) lk of 

 A(r) such that Rr k = Rr K , and denote by e E that exponent which is as 

 great as any other exponent in this set of elementary divisors, and assume 

 that j a, A | log x | ejr_ 1 is integrable vp to x — ; we can develop e K solutions 

 o/(l): 



* Cf. p. 349. 



