Exam_formula_sheet - Table of Laplace Transforms F(s =...

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Table of Laplace Transforms F ( s ) = L{ f ( t ) } = 8 0 f ( t ) e st dt, s > 0 L{ t p } = Γ ( p + 1) s p +1 , p > 1 , and L{ t n } = n ! s n +1 if n 0 is an integer L{ sin( at ) } = a s 2 + a 2 L{ cos( at ) } = s s 2 + a 2 L{ e at f ( t ) } = F ( s a ) , s > a L{ u ( t a ) f ( t a ) } = e as F ( s ) , s > a 0 L{ f ( n ) ( t ) } = s n L{ f ( t ) } s n 1 f (0) s n 2 f I (0) · · · sf ( n 2) (0) f ( n 1) (0) , n 0 L{ t n f ( t ) } = ( 1) n F ( n ) ( s ) ( 1) n d n ds n F ( s ) , n 0 L F f ( t ) t k = 8 s F ( x ) dx L F8 t 0 f ( x ) dx k = 1 s F ( s ) L{ f ( t ) g ( t ) } L F8 t 0 f ( t x ) g ( x ) dx k = F ( s ) G ( s ) , where G ( s ) = L{ g ( t ) } L{ δ ( t a ) } = e as , a 0 L{ f ( t ) } = 1 1 e ω s 8 ω 0 e st f ( t ) dt whenever f is periodic with period ω
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Summary of Fourier Series 1. The Fourier sine series of a function f de fi ned on [0 , L ] is given by 3 n =1 b n sin p n π x L Q , b n = 2 L 8 L 0 f ( x ) sin p n π x L Q dx, n 1 . 2. The Fourier cosine series of a function f de fi ned on [0 , L ] is given by a 0 2 + 3 n =1 a n cos p n π x L Q , a n = 2 L 8 L 0 f ( x ) cos p n π x L Q dx, n 0 .
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