DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS 



I 1 



(log a:)' 



G h (x) <-(t 1+ l) 



357 

 (0 < x < v < 1) 



X t\ X 



G h+ i (x) = J ±G h (x) dx = f l -G k (x) dx+ f i G h (x) dx 



C c r\ 



X x X 



\J l -G tl (x)dx\ < f 1 - | G tl (x)\ \dx\ < \(t, + l)jl!*£h\dz\ 



v " ' (0 < x < v ) 



« n l(loga:y.+i--(log^+i| 



€ 



2 



h+ 1 

 /log *7V'+ 



= -|logxKi+ 1 1 — . 



Now choose X in the interval < x < 77 so that : 



Then: 

 (log *)'.+! ^ +1 {X) 



^7+1 / - G k ( x ) dx 



(log a:) 



<I 



= ^|logx|'i+ 1 . 



(0< z< X). 



(log a-; 



^JV* (X) ** + | (log»)*4* /^ ^ dx 



< e, 



and 



(0 < z < X), 



limit 7i \7+I G *\+* ( x ) = °- 



a^o (loga-)^ 1 lT 



Therefore VI. is true for all values of t. 



Lemma. VII. If /? is a continuous function of x in the interval 

 < x < c ^ 1 , <rarf sr<cA £/m£ zV z's wo£ greater in absolute value than the 

 co?istant N, and if G t is defined as in VI, then : 



(42) 

 For : 



T X 



t integrations 



\G l (x)\<Jl\d,\..J\N\dx\<NJ l -\dx\...j l -\dx\ 



c c c c 



X 



< N\ 



