﻿CONTtNUOUS i''UNCTIONS 



To evaluate B: When a is of the form dii + l, (n being an in- 

 teger) , 



sin a>*' + >'x'n=- sin a" {2a~- i ) - ^ - (-i )«, ^ 



{since x' = -^), 



sin a"' + '^^Xa7t=sin a^^ixi +a)7t, 



(since Xi=a'*^Xo -a), 

 — sin a*'x 1 7t cos a" an + cos a^*Xi n sin a^an 

 - (—ly^ sin a'^XiTt, 

 (since sin a^Hxn^O, and cos a'*^a7i=(-iy) . 



Substituting these values in B, we have 



^ n [—, {sin a>'^ + "x' 7t- sin a^*^ ^ *^Xf,n)A 



0 X—Xo / J 



= 2,„ (-7— - (-^)« sin a^'XiTt] ) 



(I X 



. , «^ 1 + sin a'^XiTi 



= {-iy (ab)'''^n h — 



0 1 



Each term of the last summation is positive, and the first is 

 since sin XjTt is positive. 



But // ^^—4 ^ii^ ig Q^igQ value of the derivative at Xq. 



X =Xq x" — Xo 



To find the value of this fraction we break it up into two series 

 as above, and call them A and B. 

 A evaluates exactly as before. 



To evaluate B: 



sin a"' + *''x"7t = sin [a''(2a+3) ~] = — ( — i)" . 



This shows that the numerator remains unchanged, while the de- 

 nominator has the sign opposite to that of X^ Xf\. 

 Therefore, when a = 4^ + 1, and o < h < 1, 



x' — Xo ^ ' ^ '5 ah — 1 



where rj > 1, and — 1 < s < 1. 



where yj^ > 1, and — 1 < 6-^ < 1. 



