By I. J. Schwatt
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Extra resources for An Introduction to the Operations with Series
A („)(*- a) ,B - < 84 > < 85 > < 86 > (135), a=0 = 2 ( - iyig( W^-ji, ^( - V'^^Z therefore Applying (85) to sees = 4. /- we obtain, 1)n22 X 2n by means n j ( -^(jb-a) of Ch. ) ? a y = cotx. I. if 1 ( - 1)a G-a) a2B - (87) jg. fix" y=i+^TTi < 88 > DERIVATIVES OF TRIGONOMETRIC FUNCTIONS But, by Ch. I. (83), -^ k * d«yt dx (^gS(-ir(a)an 1 uk N W (w + i sin a;)* " 1 (cos x - l)^ 1 ~ (90) l; ( 1 2*+ 1 i + 1 sin 2 x sin*" 1 z fc = ~ 9^i cosec2a: ( 1 -icotx)*- 1 and where 2 ^'^^b) 00^ w . x g £g (-l) a (Ja^W 2 ,.
Applying (33) to (29) and We shall express cos Now taking the ^ and ^ sin sum of (27) C03 and (35) to (30) gives the values of and (35) 8 and Sv also as summations. (28), T=2(72r[( we have 1+i''" + ( 1 -^ < 36 > their difference gives [2=1] p] C+0-=g(-D*(a + i But and (1 -i)»=g(- then by means of (38) and g (-D'G^O ^ffl-^jV (39), we tt7T l) obtain from (36) and (37) 1 COS and sin Applying mr cos-^ and (40) to (29) way we In a similar . nir + 1 sin -j = 1 2 1 (41) to (30) gives (21) obtain by means of .
We shall first find the expansions of sin If (i) y = Bin ar, then = 8m dx^ \ x+ a: and coax. ~J and ~ dn y~\ mr =sin dxn J x =o 2 Now — . = 0, if n is even, -(-1)13, Writing 2w+ 1 for n if n is odd. is even, and since in (2), • a;2n+l therefore If (ii) & y = cos x, then =C0S (* + dn y~] T> nir and L2 = ( - 1) f-1 = 0, if w is if -', w odd. 2n We then obtain 2/ = 2(-l)* (*•)' 2. (i) To y = tan Given find and the expansion Now and of y= ~= dxn 2t-rdx n y. 2i ,2ix + =-, u+V 28 -I, I where u = e 2ix .
An Introduction to the Operations with Series by I. J. Schwatt