328 UNIVERSITY OF VIRGINIA PUBLICATIONS 



Therefore the product in (21) converges to 1 as a; goes to 0. Hence what- 

 ever be the way m which x converges to the limit of 4'{x)/x is k. 



10. Let T{x)=\l/{x)/ip(x), then as x increases from to Jir the function 

 T{x) continuously increases from to 1. When x converges to the hmit 

 of 



T(x) ^ i {x) 1 



X X ip{x) 



is clearly fc. Whatever be the real number m the limit for m = co of 



_ 1 ^ 2m \ x/m / 



is e° = 1. For any positive integer m the defining law (e''')™ = e'"'' gives 



[ip{x) -\- i^p{x)] ™ = ip{mx) + i\p{mx). 



The left side can be written in m + 1 terms by the binomial formula. On 

 equating the real and the imaginary components 



,p{mx) = v^{x) [1 - C^,2T\x) + C^,iT\x) - . . . ], 

 rPimx) = ip^ix) ImTix) - Crr„zTKx) + . . . ]. 



Inthe first of these put x/m for x, thence <p(x)/(p'°-[ — 1 is equal to 



\m/ 



\ m) 2\\ x/m J \ m] \ m] \ mj 4 ! \ xjm / 



Each term of this sum, when m is sufficiently great, is less than the corre- 

 sponding term of the absolutely convergent infinite series. 



1- ^ii:2 + ^i^4_ . . . 



2! 4! 



in which K is some assigned positive number greater than fc. Therefore 

 when m is infinite the sum is absolutely convergent and 



^W=l-^ + ^ (20) 



