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



do ,,ot involve . (p, .) give p sinh p, ; and each of the remaining terms is easily shown 



to be of the form ft (p, 5 ) sinh p*. 



Recalling that 8 (X) is independent of *, we thus see that 



when, in analogy with the notation explained in 6, a ( P ) is a shorthand symbol for 

 " Jtion of p* which is limited for values of the argument that are greater than a 

 certai i fixed positive number." If, in a similar way, we use (p, , t) to denot 



:;;;,,,;,, ,' " . -d < ** ***** * ** /> **. ^ **> 



certain positive number, and of . and t that lie both m the closed interval (0, ,), tht, 

 formula, of this and the preceding paragraph, when supplied m the expressions for 



K X (M) obtained in 4 > 8 ive 



where . , A 



r l s t ) = COsh pS COsh p (77-Q ( s ^ ^ 



p sinh pir 



_ cosh p (TT-.S-) cosh pt / ^ ^ 

 p sinh pir 



8. Let /(*) be any function which has a Lebesgue integral in the interval (0, IT). 

 Then from (13) we obtain 



k (, t)-T,(s, t)]f(t)dt = pV(.s, t)a(p, s, t)f(t)dt. 



Now when s ^. t we have 



, A _ cosh P(B g + Q + cosh p(irst) . 

 P r A.*J 2 sinh pir 



hence, recalling that I\ (x, ) is a symmetric function of s and t, we see that 



pl\ (.s-, A) ^ coth pn-, 

 for 2 * S ir, : < < IT. 



Sine.. c,,th pTT, considered as a function of p, is limited in any range which does not 

 include points within an arbitrarily small distance from the origin, it appears that, 

 for ^ s S n, : t S TT, and for all values of p greater than any assigned positive 

 number, pl\ (.,<) is limited. The same remark therefore applies to the function 

 pF A (s, t) a. (p, s, t). 



Again we have 



Hm pr x (.s, t) = o (**t) =1 (* = t). 



