USE OF DEMOIVRE'S FUNCTIOjST. 4S 



Wheu k = 0, J' 1> Y + 1, 2, |- + 2, &c. 



.X- =],-], ]'-— — F— F— — F— &c. 



?t , 71 , n , n , 



by the application of (5.) 



Then the linear factors are 



(.-1) „+,)(.- f'^) (x+f^) (.-f^) (.+ riï) ... 



And by multiplication the quadratic factors are — 



or reducing by means of (6), 



, / 2,71 \ / 47r \ 



C-x--— 1) ( X-—2X cos — + i; ( xr—2x cos — + l) • ■ • 



In a similar manner are factored the cases where n is odd. and where — 1 is written for 

 + 1. 



Ex. 2. To sura to n terms the series 



Co.s a + cos ia -\- B) -\- cos (« + 26') + ... cos {n-\-n^\.H) ; 

 And Sin «•4-siii {a + 0) -\- sin {a-\--lO) + . . . sin {n -\-n—V.fl). 



Denote the sum of the first by C and of the second by S. 



Then, C + iS = Va + V(a + 0) + F(a + 26) + . . . l\a + »^^.^) 



= Ya Î 1 + T'l^ + V'H + T'e 4- . . . F" ~\ I 

 Va{\ — rne) 

 - 1— F^ 

 by making use of (1) and summing 1 + Fi^ + ( Vfty- + ... as a geometric series. 



The realizing factor for the denominator being 1— F"V, we multiply both parts of the 

 fraction by this and obtain 



Va - r(a + nH) - yja-d) + V(^a-{-n- 1 . 0) : 



2- {vo + r-^â) 



which, upon reducing the numerator by (12) and the denominator by (7) gives — 

 ^ , .„ Sin jne.Vija+^l^ie) 



and equating real, and imaginary parts, we have finally. 



Sin ^n d cos (a-\-i n — 1 B) 

 SinJW " 



Sin J n é* sin (a+J n — 1 B) 

 ^= Sine B 



Ex. 3. To sum to infinity the series — 



Co.s a + X cos (_a+B) + x^ cos {a + 2B) + . . 

 And Sin a + .»• sin («4-^) + x- .^in («+2^) + . . , 



