formulas - n-1 n x 2 n(2 n = 1-x 2 2 x 4 24 ·· sin x =...

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Laplace Transforms DEFINITION. L{ f ( t ) } = R 0 e - st f ( t ) dt = F ( s ) . THEOREM. L{ c 1 y 1 + c 2 y 2 } = c 1 L{ y 1 } + c 2 L{ y 2 } . THEOREM. L{ e at f ( t ) } = F ( s - a ) . THEOREM. (a) L{ f 0 ( t ) } = sF ( s ) - f (0) . (b) L{ f 00 ( t ) } = s L{ f 0 } - f 0 (0) = s 2 F ( s ) - sf (0) - f 0 (0) . THEOREM. L{ tf ( t ) } = - F 0 ( s ) . FACT. Γ( 1 2 ) = π. DEFINITION. The step function u c ( t ) = ( 0 , if t < c 1 , if t c. THEOREM. L{ u c ( t ) f ( t - c ) } = e - sc F ( s ) . DEFINITION. δ c ( t ) = u 0 c ( t ) = ( 0 , if t 6 = c , if t = c. THEOREM. L{ δ c ( t ) f ( t ) } = e - sc f ( c ) . FACTS L{ t n } = n ! s n +1 ; L{ e at } = 1 s - a . L{ cosbt } = s s 2 + b 2 ; L{ sinbt } = b s 2 + b 2 . Power Series f ( x ) = n =0 a n ( x - x 0 ) n , where a n = f ( n ) ( x 0 ). f 0 ( x ) - n na n ( x - x 0 ) n - 1 , f 00 ( x ) - n n ( n - 1) a n ( x - x 0 ) n - 2 , etc. e x = n x n n ! = 1 + x + x 2 2 + x 3 6 + ··· cos x =
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Unformatted text preview: n (-1) n x 2 n (2 n )! = 1-x 2 2 + x 4 24 + ··· sin x = ∑ n (-1) n x 2 n +1 (2 n +1)! = x-x 3 6 + x 5 120 + ··· 1 1-x = ∑ n x n = 1 + x + x 2 + ··· ln (1 + x ) = ∑ ∞ n =1 (-1) n +1 x n n = x-x 2 2 + x 3 3- ··· 1...
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This note was uploaded on 12/15/2011 for the course MAP 2302 taught by Professor Tuncer during the Spring '08 term at University of Florida.

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