124 DR J- MERGER: STURM-LIOUVILLE SERIES OF NORMAL 



We have thus established the theorem : 



If the system of normal functions fc(), ^(), ., <M), * * <Aa <Ae upper 

 limit of |fc(*)| n ( a > 6 ) " few <Aan a ^ erf ww6er, /or a/Z positive integral values 

 of n, then 





provided that f(s) has a Lebesgue integral in (a, b). 



It will be seen at once that particular systems of normal functions for the interval 

 (0, IT) are defined by 



(i) V.() = V cos(n-l)* (n> 1), 



(-!), 



(ii) ^ (.v) = /\/| sin (n-J) s (w 2= 1), 

 (iii) ^ n () = V/- cos (n-J) s (n > 1), 



77 



(iv) \II H (S) = -Y/-sinns (w >: 1).* 







As each system satisfies the requirement of the theorem just stated, we see 

 that : 



If f(s) is a function which has a Lebesgue integral in (0, TT), then 



{' f (t)* [n %nt dt 



J(/ ^ ' COS 2 



tends to the limit zero as the positive integer n increases indefinitely. 



By applying this to the function which is equal to/() in an interval (y, 8) of (0, IT), 

 and is zero elsewhere, we see that 



provided that /() has a Lebesgue integral in (y, S).t 



7. Using the notation of 3, 4, let us suppose that the normal functions of 



* <y. iv., 9, n, 12. 



rhe limitations that (y 8) should be within (0, ,), and that should be integral may be removed 



th e " tTo! ?- ^ e " Undated i8 8UffiCient f r ^ ^*' A " ^ernative'proof of the 

 (1907), pp. 674-6. 



ses. n aernatve proof o e 

 n ,u .most general form will be found in HOBSON'S <Theorv of Functions of a Red Variable/ 



