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# chpp09 - 9 Laplace C Laplace C ~ ^ C 3 n 9 k 2 A^ 9.1...

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Unformatted text preview: 9 Laplace C Laplace C ~ ^ C , 3 ! n 9 k 2 A^ . 9.1 Laplace C Laplace C C . Laplace C X J F ( p ) = Z ∞ e- pt f ( t )d t (1) p t , p E . K F ( p ) f ( t ) Laplace C . f ( t ) F ( p ) O Laplace C . P F ( p ) = L { f ( t ) } f ( t ) = L- 1 { F ( p ) } Note : f ( t ) A n f ( t ) η ( t ) . η ( t ) Heaviside ( F ) η ( t ) = 1 , t > , t < (2) Laplace C 3 ^ Laplace C 3 ^ Z ∞ e- pt f ( t )d t , p , . 3 , b f ( t ) v 1. f ( t ) f ( t ) 3 m ≤ t < ∞ Y , 3 ? k m S Y : 8 k ; 2. f ( t ) kk O , = 3 M > 9 s ( O ), u ? t ≥ 0, | f ( t ) | < M e Bt (3) K f ( t ) Laplace C 3 Re p > s 3 . 3 d S , F ( p ) . Laplace C 3 ^ . K U v ^ . , , X J B 3 , K , ' B ? w , ^ . B e . I , P s . Example 9.1 f ( t ) = 1 Solution Re( p ) > , L { 1 } = Z ∞ e- pt d t =- 1 p e- pt ∞ = 1 p I s = 0. Example 9.2 f ( t ) = e αt Solution Re p > Re α , L e αt = Z ∞ e- pt e αt d t = 1 p- α I s = Re α . 1 Theorem 9.1. e f ( t ) v Laplace C 3 ^ , K lim Re p → + ∞ F ( p ) = 0 Proof s = Re p . | F ( p ) | ≤ Z ∞ e- pt f ( t ) d t ≤ M Z ∞ e- ( s- B ) t d t = M s- B Re p = s → + ∞ , F ( p ) → 0. 9.2 Laplace C 5 5 1 Laplace C 5 C . = e F 1 ( p ) = L { f 1 ( t ) } , F 2 ( p ) = L { f 1 ( t ) } , K L { α 1 f 1 ( t ) + α 2 f 2 ( t ) } = α 1 L { f 1 ( t ) } + α 2 L { f 2 ( t ) } = α 1 F 1 ( p ) + α 2 F 2 ( p ) (4) 5 N l Laplace C . 5 , = L { sin ωt } = L e i ωt- e- i ωt 2i = 1 2i 1 p- i ω- 1 p + i ω = ω p 2 + ω 2 (5) L { cos ωt } = L e i ωt + e- i ωt 2 = 1 2 1 p- i ω + 1 p + i ω = p p 2 + ω 2 (6) 5 2 L { f ( t ) } = F ( p ) n L { f ( t- τ ) } = e- pτ F ( p ) , τ > (7) q 5 L { f ( at ) } = 1 a F p a , a > (8) n L e p t f ( t ) = F ( p- p ) (9) 5 3 Laplace C . f ( t ) 9 f ( t ) v Laplace C 3 ^ , L { f ( t ) } = F ( p ), K L { f ( t ) } = pF ( p )- f (0) (10) Proof , = Z ∞ f ( t )e- pt d t = f ( t )e- pt ∞ + p Z ∞ f ( t )e- pt d t 2 , f ( t ), f ( t ), ..., f ( n ) ( t ) v Laplace C 3 ^ , L { f ( t ) } = F ( p ), K L { f 00 ( t ) } = p L { f ( t ) } - f (0) = p 2 F ( p )- pf (0)- f (0) (11a) L n f (3) ( t ) o = p 3 F ( p )- p 2 f (0)- pf (0)- f 00 (0) (11b) L n f ( n ) ( t ) o = p n F ( p )- p n- 1 f (0)- p n- 2 f (0) (11c)- ··· - f ( n- 1) (0) (11d)...
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chpp09 - 9 Laplace C Laplace C ~ ^ C 3 n 9 k 2 A^ 9.1...

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