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# Tables - Mathematical Tables Composed by Vincent Verdult...

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Unformatted text preview: Mathematical Tables Composed by Vincent Verdult Department of Electrical Engineering Delft University of Technology e-mail: [email protected] November, 1997 Contents 1 Trigonometric Identities 2 2 Trigonometric Functions 3 3 Hyperbolic Functions 3 4 Series 4 5 Inequalities 6 6 Differential and Integral Calculus 6 7 Integral Table 8 8 Standard Limits 11 9 Convolutions 12 10 Dirac Delta Function 12 11 Fourier Transform 13 12 Laplace Transform 21 13 z-Transform 25 1 Trigonometric Identities sin( x ± π 2 ) = ± cos x cos( x ± π 2 ) = ∓ sin x sin( x ± y ) = sin x cos y ± cos x sin y cos( x ± y ) = cos x cos y ∓ sin x sin y tan( x ± y ) = tan x ± tan y 1 ∓ tan x tan y 2sin x sin y = cos( x- y )- cos( x + y ) 2cos x cos y = cos( x- y ) + cos( x + y ) 2sin x cos y = sin( x- y ) + sin( x + y ) sin x + sin y = 2sin ( x + y ) 2 cos ( x- y ) 2 sin x- sin y = 2cos ( x + y ) 2 sin ( x- y ) 2 cos x + cos y = 2cos ( x + y ) 2 cos ( x- y ) 2 cos x- cos y =- 2sin ( x + y ) 2 sin ( x- y ) 2 tan x ± tan y = sin( x ± y ) cos x cos y sin2 x = 2sin x cos y cos2 x = cos 2 x- sin 2 x tan2 x = 2tan x 1- tan 2 x 2sin 2 x = 1- cos2 x 2cos 2 x = 1 + cos2 x 2 4sin 3 x = 3sin x- sin3 x 4cos 3 x = 3cos x + cos3 x 8sin 4 x = 3- 4cos2 x + cos4 x 8cos 4 x = 3 + 4cos2 x + cos4 x a cos x- b sin x = r cos( x + θ ) where r = a 2 + b 2 θ = arctan b a a = r cos θ b = r sin θ 2 Trigonometric Functions sin x = e jx- e- jx 2 j cos x = e jx + e- jx 2 tan x = sin x cos x cos 2 x + sin 2 x = 1 e ± jx = cos x ± j sin x 3 Hyperbolic Functions sinh x = e x- e- x 2 cosh x = e x + e- x 2 tanh x = sinh x cosh x cosh 2 x- sinh 2 x = 1 e ± x = cosh x ± sinh x arcsinh x = ln( x + x 2 + 1) arccosh x = ln( x + x 2- 1) , x ≥ 1 arctanh x = 1 2 ln 1 + x 1- x , | x | < 1 3 4 Series Series Expansions f ( x + a ) = ∞ n =0 x n n ! f ( n ) ( a ) = f ( a ) + x 1! f ( a ) + x 2! f ( a ) + ... Taylor’s series e x = ∞ n =0 x n n ! = 1 + x 1! + x 2 2! + x 3 3! + ..., | x | < ∞ sin x = ∞ n =0 (- 1) n x 2 n +1 (2 n + 1)! = x- x 3 3! + x 5 5!- x 7 7! + ..., | x | < ∞ cos x = ∞ n =0 (- 1) n x 2 n (2 n )! = 1- x 2 2! + x 4 4!- x 6 6! + ..., | x | < ∞ sinh x = ∞ n =0 x 2 n +1 (2 n + 1)! = x + x 3 3! + x 5 5! + x 7 7! + ..., | x | < ∞ cosh x = ∞ n =0 x 2 n (2 n )! = 1 + x 2 2! + x 4 4! + x 6 6! + ..., | x | < ∞ arcsin x = ∞ n =0 (2 n )! 2 2 n ( n !) 2 x 2 n +1 2 n + 1 = x + 1 2 x 3 3 + 3 8 x 5 5 + ..., | x | ≤ 1 arccos x = π 2- arcsin x = π 2- ∞ n =0 (2 n )! 2 2 n ( n !) 2 x 2 n +1 2 n + 1 , | x | ≤ 1 arctan x = ∞ n =0 (- 1) n x 2 n +1 2 n + 1 = x- x 3 3 + x 5 5- x 7 7 + ..., | x | ≤ 1 arctanh x = ∞ n =0 x 2 n +1 2 n + 1 = x + x 3 3 + x 5 5 + x 7 7 + ..., | x | < 1 ln(1 + x ) = ∞ n =1 (- 1) n +1 x n n = x- x 2 2 + x 3 3- x 4 4 + ..., | x | ≤ 1 1 1- x = ∞ n =0 x n = 1 + x + x 2 + x 3 + ..., | x | < 1 (1 + x ) a = ∞ n =0 a n x n = 1 + a 1 x + a 2 x 2 + a 3 x 3 + ..., | x | < 1 where a k = a · ( a- 1) · ( a- 2) ··· ( a- k + 1) k !...
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Tables - Mathematical Tables Composed by Vincent Verdult...

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