Section III., 1896. [ 167 ] Tuans. R. S. C. 



X. — Symbolic Use of Demoivre's Theorem. 



By Professor Dupuis. 



(Read May 20, 1895.) 



At the meeting in 1891 of this society I presented a paper upon the 

 Symbolic Use of Denioivre's Theorem, with illustrations and examples. 

 The present paper consists of additional illustrations of the application of 

 this method of using- the theorem. 



1. To put (a + lb) ^ + '"^ in the form A + iB. 

 Let a -{- ib= r V/3. 



Then (a + ^7.) ^ + ''^ =(/■ r/3) « + ^<^ = ;.«./^. Yep. Vid/3. 



But /^ = e^^-^^ = Vd. Ir: and Vidf3 = e"^/^- 



where r = Ja'^ -\- b', and cos /3 = -• 



n 



2. To sum the series 2 sin -na. 



1 



2i sin a =iV — V~^ by separating the operative symbol. 

 ... sin ^-a=~-\(^Y-JrV-'^), 



.-. 1 sin ^wo' = I _ ^ I V' + V*-\- . . V''\ + V-' + F-' + . . F-^4 

 n irF2n+_2y^2 ^_2n-2^-2| 

 = 2~"4l V' — l + ~~V-' — l J 5 

 which, being reduced and reahzed (see former article), gives 



n cos (n + 1)'^ sin na 

 2 2 sin a 



B. To sum the series 



"^^^^sin-é^ . , -.^^ , '^ = «'sin"^ ^ ■ 



denote the generating function of the first by C, and of the second by S, 

 Then 



G-\-iS^ F|l + fT + ^2T + ]=Ve,,. 



= Fé'-^^e^^' = e*«F6'Fsin2^ 

 ^gSin0cos(^-p^J^_^^i^2^j 



= e «'° '^ ^« ' jcos (^ + sin 2^) + . sin (^ + sin2^) ( 



