Signal Processing and Linear Systems-B.P.Lathi copy

# 7 miscellaneous b7 5 complex numbers b1 1 lhopitals

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Unformatted text preview: . , . + - + ... = 1+x x 3 x5 t an x = x x3 2x 5 - = H e jX x = ~[ejX - e- jX ] sin {x ±~) = ± cos x 2 sin x cos x = sin 2x x 2 < 7(2/4 sin 2 x 17x 7 - - + .. , 315 2 x 2 < 7(2/4 + cos2 x n (n + x) n = 1 + n x + - -!-1) 2 + n {n x 2 l){n - 2) 3 ( n) k x + ... + x 3! k I xl«:l =1 2 cos x - sin x = cos 2x cos 2 x = ~(1 I=::ll+nx + e - jX ] cos {x ±~) = ' Fsin x ... 17x 7 2x 5 +15 x sin + -3 + -15 + -15 +... 3 x3 t anh x = x - 3 = cos x ± j sin x cos x7 sin x = x - - + - - - + ... 3! 5! 7! x2 x4 x6 x8 c osx=l- -+---+-2! 4! 6! 8! (1 = r lr2 ej (9 1+92) .. + ... + x n + cos 2x) sin x = ~(1- cos 2x) 2 cos 3 x = 1 (3cos x + cos 3x) sin 3 x = i {3sin x - sin 3 x) 1 2 3 - -=l+x+x + x + ... Ixl < I -x 1 sin (x ± y) = sin x cos y ± cos x sin y cos ( x ± y) = cos x cos y 'F sin x s in y B.7-4 Sums ~ t an (x ± y) = l r m _ r k + _1 ~ r-l + y)] + cos (x + y)] sin x cos y = H sin(x - y) + sin {x + y)] a cos x + bsin x = C cos ( x + IJ) sin x sin y = ~[cos (x - y) - cos (x rfl m =O cos x cos y = ~[cos (x - y) N L t an x ± t an y 1 'F t an x t an y ----~ r f1 m =M a fb in which C = v 'a 2 + b2 a nd 1 IJ = t an- ( :b) 48 B.7-7 Background Indefinite Integrals J= J J . J. J J J J a~ J J. . J. J U dv uv - B.7-8 J J cos a x d x sin 2 ax sm ax dx = - - - - - = ;1 s m a x ;. 2 x cos a xdx = 2 4a x s in a x d x = sin 2 ax + --4a x c os a x d x = a13 ( 2ax cos a x - 2 sin a x bd s maxsm x x = [ coS(a-b)x ( ) 2a- b cos a x cos bx d x = s in(a-b)x ( ) 2 a-b + d a dx V l- a 2x 2 - (sin - 1 a x) = -;===;;F~ d -a - (cos- 1 a x) = - r.;==;;==;r 22 dx x a 2 x 2 cos a x) VI - d - (tan- 1 a x) = dx a 22 1+a x dx + a 2x 2 sin a x) B.7-9 Some Useful Constants 7r " " bd d a d x t an a x = cos2 a x 1 d loge - log(ax) = - s in(a-b)x s in(a+b)x 2 (a-b) 2(a+b) s m a x cos x x = - dx d 1 - In(ax) = dx x + 2 cos a x - 2 d . - cos a x = - asm a x dx v2 v d xn _ = nx n dx + a x sin a x) ( 2ax sin a x d. - sm a x = a cos a x dx x s in a x d x = : 2 (sin a x - ax cos a x) x c os a x d x = a12 (cos a x d _ a bx = b (lna)a bx dx d () v4!!: _~=dx - uih!. dx j (x)g(x)dx x 2 Differentiation Table d dv du - (uv)=u-+vdx dx dx J 1 s m a x d x = - ;; cos a x 2 49 d d du d xf(u) = d u f (u) d x v du f (x)g(x)dx = f (x)g(x) - 2 B.7 Miscellaneous + C OS(a+b)x] ) 2(a + b 3.1415926535 e "" 2.7182818284 i "" 0.3678794411 s in(a+b)x ( 2 a+b) loglO 2 = 0.30103 loglO 3 = 0.47712 B.7-10 Solution o f Quadratic and Cubic Equations Any q uadratic e quation can be reduced t o t he form J J + J__ J e e ax ax x2 ax 2 + bx + c = 0 x s in b xdx = a 2e: b2 (asin bx - bcos b x) cos bx d x = a 2e:b 2 (a cos bx 1 _ d x = ~ t an - 1 a2 - x 2 dx = 2 x +a a :. a ~ In(x 2 + a 2 ) 2 + b sin b x) T he solution of this equation is p rovided by x= -b±~ 2a ax 50 Background A general c ubic equation y3 + p y2 + qy +r = 0 may be reduced t o t he d epressed c ubic form x 3 + ax + b = 0 by s ubstituting y=x-~ Introduction t o Signals and S ystems T his yields b = f.r(2 p 3 - 9pq + 27r) Now let A_ - 3 b ~ -2+Y4"+Zf, B=V...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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