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

7 miscellaneous b7 5 complex numbers b1 1 lhopitals

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online