MATH 202 Other, Spring 2008

Table of Laplace Transforms L {f(t)} = � ∞ 0 e−stf(t) dt = F(s) f(t) L {f(t)} = F(s) f(t) L {f(t)} = F(s) 1 1 s eat 1 s − a tn n! sn+1 eattn n! 1 (s − a)n+1 sin ωt ω s2 + ω2 cos ωt s s2 + ω2 eat sin ωt ω (s − a)2 + ω2 eat cos ωt s − a (s − a)2 + ω2 t sin ωt 2ωs (s2 + ω2)2 t cos ωt s2 − ω2 (s2 + ω2)2 sin ωt − ωt cos ωt 2ω3 1 (s2 + ω2)2 U(t − a) e−as s δ(t) 1 δ(t − a) e−as L {αf(t) + βg(t)} = αL {f(t)} + βL {g(t)} L {eatf(t)} = F(s − a) L {tnf(t)} = (−1)n dn dsn F(s) L −1{e−asF(s)} = f(t − a) U(t − a) L {f(t) U(t − a)} = e−asL {f(t + a)} L {f′(t)} = sF(s) − f(0) L {f′′(t)} = s2F(s) − sf(0) − f′(0) L {f(n)(t)} = snF(s) − sn−1f(0) − · · · − f(n−1)(0) f(t) ∗ g(t) = � t 0 f(θ)g(t − θ) dθ =⇒ L {f(t) ∗ g(t)} = F(s)G(s) L �� t 0 f(θ) dθ � = F(s) s f(t) has period T =⇒ L {f(t)} = 1 1 − e−sT � T 0 e−stf(t) dt Gilles Cazelais. Typeset with LATEX on April 20, 2006.