Oscillatino' Dirichlet's Inteo-rals. 



89 



Tlie integrals J'i(n), J-ln) and J{ii) are all convergent, if 



f> < xa' 



as x->0. Introducing the above expressing of {> and ö-, this con- 

 dition becomes 



which is nothing but our h3'^pot]iesis. 



Thus, El and e,, having the above properties, Lemma (S of 

 Part I may be applied to our integrals, and we obtain 



Ji{n) r^ I X ^cos('Ha:' + — J dx, 

 JJn) (O / X'" sin ( nx -\- — \ dx 



(74) 



as M.->oo , and hence we l^ave 



J^{n) r^ A-*^+-M*-^ COS {1ahi^-+\-) (-^<^j<:2), 



Jin) r^ -*a-^^+M^-^ sin (2aM + i-) {-^<p<'l). 



Hence we have 



J{n) r^ i--ia-*^+'e-^'7t^*-^exp[-(2aM -\pn + \~)i], 

 (_i<p<2). 



51- (ii) The case in which 0<:g<;l, « = (1 + ^)^, ^xl + q. 

 In this case 



a + ^q- = I' + q-, 



and this lies between hr. and |-, since < g < 1. By (70), we 

 have : 



If g ^ h 



</>(d) = -|L COS qr: + ^p- + Oiä'-') ; 



