Math 242 (Section 3)  University of South Carolina
D. Meade  Spring 1998

Table of Laplace Transforms

y(t) = L^{−1}[Y] Y (s) = L[y] y(t) = L^{−1}[Y] Y (s) = L[y]
e^{at}  \frac{1}{s−a}, s > a  t^{n}  \frac{n!}{s^{n+1}} , s > 0
sin(ωt)  \frac{ω}{s^{2}+ω^{2}}  cos(ωt)  \frac{s}{s^{2}+ω^{2}}
e^{at} sin(ωt)  \frac{ω}{(s−a)^{2}+ω^{2}}  e^{at} cos(ωt)  \frac{s−a}{(s−a)^{2}+ω^{2}}
t sin(ωt)  \frac{2ωs}{(s^{2}+ω^{2})^{2}}  t cos(ωt)  \frac{s^{2}−ω^{2}}{(s^{2}+ω^{2})^{2}}
u_{a}(t)  \frac{e^{−as}}{s} , s > 0  δ_{a}(t)  e^{−as}
\frac{dy}{dt}  sY (s) − y(0) y(t) + w(t) Y (s) + W (s)
αy(t)  αY (s)  u_{a}(t)y(t − a) e^{−as}Y (s)
e^{at}y(t)  Y (s − a)

Equilibrium Analysis for Linear Systems