496 The Rev. Brice Bronwin on some Definite Integrals. 

 changing the signs of the alternate results, and adding, be- 

 cause u~ z ~ l =w z + I , we have 



,z+i - 



js-2 



*-4 



^ l =y z _«*-' + «*-*_ + y 



,n J m,n J m,n J m,n u 



m,n 



We suppose z an even integer. We may add and subtract 



— v~ z ~ 2 + v~ z ~ i — , &c. continued till they become zero. 



Then putting for these quantities their values from (1.) and 

 («.), there results 



»?,» 



_*_4)»-l + ,..} 



n-1 



;...,) 



+ 2 ^4^ {A(w -*- 2r ' 1 -^ 



(D 



A=l, A, 



1+y, A s 



m m(m — 1) 



1+ T + iTs ' &c * 



If we make z + m =p + 2 1, the first series of (D.) may be 



put under the form BA < p n - 1 — B 1 A f - ] j> M " 1 + ••• ± ^t p n ~ l - 

 Expanding these differences, we find by comparison B = 1, 

 B x = 2, B 2 = 2 2 , &c. The second series may be transformed 

 in like manner. As particular examples, reversing the series, 

 we have 



/ 



dx . cos(2r— \)x 



-(sin#) 2r - 



_(-D 



i-i. 



n-\ 



cos# 

 AO"- 



f 



2P(w-l) 



dx cos(2r- l)x 



> 3in *) cos* 



1 A2 «-1 



2 2 



-'o»-n 



,2,-1 J. 



Al w_1 , A 2 l w_1 



•...+ 



A 8r-2 1 n-l'| 



(13.) 



, (14.) 



'2P(n— 1)1* 2 2 2 *" ' 2 2r - 2 /■ 



where A0 = A 1 = Az= 2, 11= m — 2f, and 2; even. We might 

 find the same integral when n — m — (2/+1), but the formulae 

 would not be simple. 



When z is odd, the latter half of the terms have contrary 

 signs to the former, and destroy them ; since y~ z = if , &c. 

 But we may continue the series to infinity. Then 



,z+l _ o ..-2-2 



IC« "* *» 2 V" 2 - 2 - 2j/- ,; - 4 + &C 



Jm.n *s w " 



-*-4 

 m, n 



All the terms vanish after z becomes less than (— m) ; but 

 there will be a remainder +u~ p = + u p , p being ii.!inite 



— m,n — m,iv * » 



and even. Now 



