168 



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



6. It follows from (8) and (15) that the large singular values of *(*,t) may be 

 calculated from the asymptotic formula 



cos 



T. = n+ 



1 f 



a(n) 



.,2 ' 





We proceed to obtain an asymptotic formula for *.(), the normal function 



corresponding to X.+j. 



Let denote the function which u ( 1) becomes when r, is substituted for r ; then, 

 from the definition of r., it is clear that u. satisfies the pair of boundary conditions B . 

 The normal function ^ +1 (s) must therefore be a constant multiple of ; hence, since 



f [/.+, ()]' ds = 1, we must have 



t/.+i (*) = 



Now, from (6), 



(16) 



u 2 da 



5^)+8mT^[^'-^ 



while from (8) we see that 



B 



cos T.S = cos )is ( 1 + sn ns 



n 



CTT 



| /7i ' +H/ ~ Jo 



~ 



* 



a (n] 



with an analogous formula for sin r n s. It follows that 



i cos ns l 



sin ns 



l cos 2 



- 

 IT Jo 



cos 



If*. 

 u. = cos ns( 1 + - qi sm nSi 



\- (h'~ - (h'+ H')- f ?l 



Ln\ ir Jo 



From the formula just written we see that 



X2 a (,, s) 



(M B cosns) = v ' ' . 

 n 3 



Integrating between the limits and IT, it will be found that this leads to 



r %,'<& = 2 [* tt .coend8--+^j) 

 Jo Jo 2 ?r 



The function 2,, cos ns is of the form 



2 cos* tw + - j y, sin ns t cos nsi ds t + - [ft (n, s) cos 2ns + ft (n, s) sin 2ns] + " ^ S ', 

 where ft(n, s) and ft(n,s) are functions of w and s whose derivatives with respect to 



(17) 



