By I. J. Schwatt
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Extra resources for An Introduction to the Operations with Series
Where We shall cot2P+i-y, (97) n ^ next find the expansion of y = xcotx. We ("^ y= have ix + (98) X (99) J£_ j dL^&^iLo and ^ Now = _± _ T ^ c 7 (100) (101) . 1 <&» 1 _Uo . _J_-] *5T| hence dk dxk _U 0 ~ - 2" 1 dx"- 1 e* + 1 n0P5 v _W 1 ' Applying ^^iLo-Sa^tX)^- (to 1 . cot^l-EC-D-^Tl 2«-l j * w. g ^S (- 1 > a (a) 1 5. To (i) (108) y = cosec£. Given in terms of find a! "- 1- (109) powers of cosec x and cot x. Let w = sin$; then, by Ch. 1. tj(*) ( |; [J"" where ' ( parts, (113) ^ - ( 113 > (114) becomes (us) _ o~ v (" 1 >\%0 ) (- 1 P ) cot20a; &+l) cot2ma; ( - 116 > (117) .
KT)s^G)rr- ) (182) and (^-^(-^-^cr)^©^^- 1 ) -2p (183) only those values of n being admissible for which (ii) n+p is a multiple of 3. Glaisher* obtains the coefficients of the expansion of x log(l+z) form of determinants. The method used cannot, however, be conveniently applied to the expansion of the more general form in the xp y = log»(l * + *) < ' The Messenger of Mathematics, vol vl p. 50. 184 > OPERATIONS WITH SERIES 24 Now, U J°8Q + Q letting u~ p ~ k ] x =o = then, since we have by To (185) 9 15 (169), n ^ rf find w "I fc , we write A* = log fc (l+z), by (185); then by Leibnitz's theorem, Now as the first member of (187) vanishes except I^*L„=ct!
An Introduction to the Operations with Series by I. J. Schwatt