This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ) . ±...
View
Full
Document
This note was uploaded on 11/11/2011 for the course MATH 255.01 taught by Professor Kwa during the Winter '11 term at Ohio State.
 Winter '11
 Kwa
 Differential Equations, Equations

Click to edit the document details