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



1 10 Let us now suppose that < s < , If - is an arbitrarily assigned positive 

 number less than, or equal to, ., the first term of the right-hand member of (17) may 



be written 



T-T 



2 sinh 



cosh p (}[" ("" ^ ft t \ d t + P e ' 1 f(t) dt . 



J \ I V \ I 



|_Jo *~" J 



Since 



2 sinlr/jTr 'o 



2smhf>ir 



it is clear that 



2 sinh 



converges to zero, as X tends to oo. 



Again, by a simple substitution, it is easily seen that 



(' <?f(t)dt = ef [V<"/(s-0 dt ; 



Jl a Jo 



and evidently 



pe" cosh p (v-s) p f -p. /(g _ A j/ T^^ g r *-*/(*-* dt. . . (19) 

 2 sinh pjr Jo 2 Jo 



We thus prove that 



p cosh p (TT-S) f' fi - p , / . ( . ^ __ g f e -* f(8 _fi dL . . (20) 



2 sinh pir Jo 2 Jo 



The restriction a^s may now be removed, provided that the function f{s) is 

 defined for values of s outside (0, TT) in any manner consistent with the condition, 

 that f(s) should have a Lebesgue integral in every finite interval. For if a > s, 

 we have 



and the right-hand member of this has been shown to differ from the left-hand 

 member of (20) by a number which tends to zero, as X tends to oo. 



By the same method and with the same convention it may be shown that 



(21) 



Hence, from (17) and (20) we see that 

 ,(s, t)f(t) dtE^ 



for all positive values of a, and for all values of s which belong to the open interval 

 (0, ). In the paragraphs which immediately follow we shall obtain a more general 

 form for the right-hand member. 



