Oscillatinu; Diriclilet's Intei^rals. 



79 



I,(?.) ^ irriw-i'-^V^-^ cos (^vii/i - ^-) ( - 1 < a <: 2), 



J.'^O ^ h-^m-^'-^Xi^'-^ sin (o^n^jih-i-) ( - 1 < a < 2), 



S{?.) ^ ^-W-*'-^;>-^ exp {(2w-^;i-i-)/] (- 1 < a < 2\ 



which also agree witli the results obtained from Theorem \IJ, 

 only the difference being that the lower limit of ^^ is -1 instead 

 of -i. 



Thus our theorems YI and YII are verified. 



PART II 

 Coefficients of Power Series 



/. Preliminaries. 

 44- Consider a power series 



m la.,'. 



n = 



whose radius of convergence is unity, representing a function 

 f{z) which has, on the circle of convergence, one singular point 

 only at - = 1, being regular at every other point on it. 



Let ABCB be the circle of convergence of the series (60), 

 A being the point z = 1 and the centre. 

 Draw a circular arc BPD inside the circle 

 of convergence, cutting it at the points 

 B and D, with the centre at A and the C 

 radius ri < 1. \ 



Let J(;-i) denote the integral 



(61) 



I{>\) 



l-i J DPB 





where the path of integration is the arc DPB, starting at the point 

 D, turning round the point A in the clock-wise direction along 

 this arc and ending at the point B. 



