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Unformatted text preview: s > α , then (6) L [ τ a f ( t )]( s ) = eas F ( s ) for all s > α . Proof. The proof is computational: L [ τ a f ( t )]( s ) = Z ∞ est u a ( t ) f ( ta ) dt = Z ∞ a est f ( ta ) dt by the deﬁnition of u a = Z ∞ es ( τ + a ) f ( τ ) dτ by the substitution τ = ta = esa Z ∞ esτ f ( τ ) dτ = esa F ( s ) for s > α . ± Conversely, multiplying the function f by an exponential term translates its Laplace transform: Theorem 2 (Second translation theorem) . If F ( s ) = L [ f ( t )]( s ) exists for s > α , then for any real constant a , (7) L [ e at f ( t )]( s ) = F ( sa ) for all s > α + a . Proof. The conclusion follows from the existence of F ( sa ) = L [ f ( t )]( sa ) for sa > α , for F ( sa ) = Z ∞ e( sa ) t f ( t ) dt = Z ∞ est e at f ( t ) dt = L [ e at f ( t )]( s ) . ±...
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 Winter '11
 Kwa
 Differential Equations, Equations, Heaviside step function, Oliver Heaviside, KIAM HEONG KWA

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