424 PROF. A. C. DIXON ON STURM LlorVILLK HARMONIC EXPANSIONS. 



It is not clear that the two integrals 



**>(*, *,B)<fc 



tend to definite limits when R->o, but their difference does so, and, in fact, if 

 a <x < I, 



I, rx rti 



\ (t,.r,T&)dt = + of the same 



.flBX *(1,0) .wX 



All the parts of this expression tend to zero when R -> QO except 



. 0) + * Or. !)*(*. 0) > 



J(R) A* (1 



which 



d\ 



I x = ~ 2nr - 

 Jon A 



18. It is now possible to prove that tf fix) ** continuous at x and of limited total 

 fluctuation in a neighbourhood of x then 



(t,.r > li)dt ....... (11) 



Jo 



For ( I ) this holds when f(x) has a constant value ( 17) ; 



(2) The contribution to the integral from values of t not lying between x/u. may 

 be ignored, the values ic/u lying within the neighbourhood where f(x] is of limited 

 total fluctuation (HoBSON s convergence theorem) ; 



(3) By the second mean-value theorem, if f\(x) is monotone, 



where ^/*, < /u.. In the last expression the second factor is finite ( 17), while the 

 first can be made as small as we please by taking p small enough, if fi is supposed 

 continuous. The same holds for the integral from x /x to x. 



Thus if f(x) is of limited total fluctuation between X/JL and is, in fact, the sum of 

 two functions /i, / 2 which are monotone between those limits, and continuous at .r, 



2,7r/(x) =f(x) x Lim rF(,*, 



R->-oo JO 



,B>*+jr% r + p+ r i 



IJO Jl-n .'j- J*+!iJ 



f f^-,1 ,v ,-X+M rl 1 



+ + + {/^-/^}F( 



L Jo x-n Jx .V+. J 



