162 DR J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



Throughout this section we shall assume that the normal functions $ n (s) hare such 

 an order that the corresponding singular values X, increase continually with n. 



We proceed, in the first place, to obtain asymptotic formulae for ^ (*) and X B , when 



ft 1L 



n is large. Let t* be the solution of (1) which satisfies the conditions u =-- 1,-^- = h' 

 at * = 0, when p has the positive value T". Since u satisfies the equation 



[D+T > Ju = -qu, 



it is evident that 



= c, cos TS + CJ, sin TS-tP'+T 3 ]' 1 qu, 



where c, and c, are constants. Proceeding as in III., 5, we thus see that 



1 f* 

 u = c, cos TS + CJ, sin TS -- q^ sin T (s i) cZs,, 



T . 



where 7,, u, are what q, u become when s t is substituted for s ; and it is easily shown 

 that the conditions satisfied by and -r- at s = give o t = 1, c, = , 



CLS T 



as the appropriate values of the constants. It follows that 



h' If 



u = cos TS+ sin TS - I <?!! sin r (s s,) cZsj ...... (4) 



T T Je 



Denoting by M the upper limit of | u \ in the interval (0, TT), we deduce from this 



the inequality 



/ J/2\V2 ,-, r* 

 S(l+y +- \qi\ds lt 



\ T / T.'O 



which may be written 



1/2,1/3 f -, fw -1-1 



IS ( I+ S) J 1 -^!*.!*.} . | 



It follows that for values of T whose lower limit is greater than f jflfi I ds 1} u is less 



Jo 



than a fixed positive number. Using the notation of III., 6, we deduce from (4) 

 the formula 



tt-OOBrt+^ill*!. (5) 



T 



The equation (4) may be written 



= cos w l + , Ul sin 



Supplying the value of u given by (5) on the right-hand side, we obtain 



= cos T* ( 1 + - I </, sin TS t cos TS, ds l + a ( T ' s ^\ 

 T Jo r 3 J 



+ 8iuTS^/ i '-f\ 1 cOS S TS 1 rfs 1 Wf5j(^L)l ... (6) 



