﻿62 



INDIAISTA ITNIVEESITY 



■ ' 2 71 



Then if we choose a and h such that - > these two 



3 ab — 1 



ratios have opposite signs, and Avhen m increases indefinitely, i. e., 

 when ~ Xq and x" — Xq approach zero, each becomes infinite. 

 Therefore, under the conditions. 



1) a = 4N + U {N being an integer), 



2) o<h<l, 



2 71 3 



3) — > 7 ? 01' otherwise stated, ah > 1 + —n, the 



o CLO — 1 2 



00 



function f{x) = 2^, sin a'^xJl, has neither a definite finite nor in- 



o 



finite derivative. 



The nature of the graph of this function can be best studied 

 by writing the function thus: 



00 



f(x) = '^n a'^xTt = sin X7t + b sin axn + b^ sin a^xn + 



o 



. . . + b'*^ sin a'^^x 7t+ . . . 

 The graph of each term of this s animation differs from that of 

 the ordinary sine function only in wave length and amplitude. We 

 can approach as nearly as we please to the graph of the function 

 f(x) by adding the ordinates of a sufficient number of terms of 

 the summation. 



Since sin x ^ 1, and o < b < 1 , we can easily find the maximum 

 of the function, 



fix) ^ 1 ^ b + b^- + b^ + . . . + b>^ + . . = 

 The maximum given by the terms after 



b" -^ sin a"'-^ xn is 6^' + i + . . . 



1 - b 



With this data, although we cannot make a graph of the com- 

 plete function, we can enclose a region in which it must lie. 



In the accompanying drawing, a —■ 9, and b .64. 



The ordinates of the graph {A) of sin xn and {B) of b sin axn 

 are added to form the graph (C) sin xn+b sin axn. The maximum 



b^ 



given by the other terms is ^ = 1.13 + . 



The graphs D and E are formed respectively by adding and sub- 

 tracting from the ordinates of C, the maximum of the remainder, 

 1.13. It is evident that the graph of the complete function must 

 lie between these two curves. Further, since the maximum of the 

 complete function is 



