422 PROP. A. C. DIXON ON STIMfM LIOUVILLE HARMONIC EXPANSIONS. 



15. Now suppose p to l>e limited both ways, and let F(t, x, R) denot.- 



If 



'( ii) 



where p t means the value of the function p of the argument t, and the path of 

 integration in the X-plane indicated by (R) is a circle of radius R with centre at the 

 origin, R l)eing such that this path does not pass through a point where * (l, 0) = 0. 

 /) ( F (t, x, R) is a symmetric function of t, x for 



,(t,l)-^(t,V) + (x,l)}(1\ 

 = \ <j>(x, t) d\ = 0, 



.(R) 



since <j>(x, t) is a holomorphic function of X everywhere ( 12). 



It will now be proved that F (t, x, R) satisfies the conditions of HOBSON'S 

 convergence theorem (' Proc. L.M.S.,' ser. 2, vol. 6, pp. 350-1), that is 



(1) Its absolute value does not exceed a certain quantity F for all values of t, x 

 such that t ; ~ x 2t /u. and for all values of R. 



f* 



(2) F (t, x, R) dt exists for all values of a, h such that : a < 1> ^ 1 and for each 



Ja 



value of x in the interval (0, l) which does not lie between a /j. and I> + n; this 

 integral, moreover, is less than a positive number A, independent of a, b, x. 



(3) A -> when R -> o. 



1G. A change in the value of R does not affect (t, x, R) unless it changes the 

 number of zeros of ^ (l, 0) enclosed by the circle : hence we may suppose the circle to 

 cross the real axis on the positive side at a point T, where v(l, 0) is zero. Thus, at 

 the point T, V(l, 0), and therefore also *(l, 0), are //v/X, since y| 2 + Dt'| 2 /|A 

 cannot tend to zero. 



Again ( 12) 



l ,l)dx, ...... (8) 



which is limited when a is limited,* whatever .the value of /3. A distance // v/X can 

 therefore be assigned such that within that distance of T >^(l, 0) -r %/X| does not 

 approach zero, but exceeds a certain fixed quantity independent of R. Beyond that 

 distance from T on the path it has been proved already (11) that *(!,()) //X 1 * exp a, 

 so that this is now proved for the whole path. 



The numerator \[, (x, 0) ^ (t, 1 ) (f > ./) is // exp ax . exp a ( 1 /), so that the subject 

 of integration in F(/, x, R) is 



// X~ l a exp a. (.t t\ that is, X~ l a exp ( /*) at most, 

 * We still take ,/- X = a + i/i; thus a is limited for points within a distance // v/X (or </R) of T. 



