# Laplace Transform Tables

Cuthbert Nyack
f(t) = d(t), unit impulse, ® F(s) = 1
f(t) = 1(t) (unit step f(t) = 1 for t > 0), ® F(s) = 1/s

f(t) = 1(t - T) (delayed unit step f(t) = 1 for t > T), ® F(s) = e-Ts / s

f(t) = t , ® F(s) = 1/s2
f(t) = tn , ® F(s) = n! / sn+1
f(t) = e-at, ® F(s) = 1/ (s + a )
f(t) = ln t, ® F(s) = -(g + ln s)/ s
f(t) = e-a(t - T), f(t) = 0 for t < T, ® F(s) = e-Ts/ (s + a )
f(t) = te-at, ® F(s) = 1/ (s + a)2
f(t) = tne-at, ® F(s) = n! / (s + a)n+1
f(t) = sin wt , ® F(s) = w / (s2 + w2)
f(t) = cos wt , ® F(s) = s / (s2 + w2)
f(t) = sinh wt , ® F(s) = w / (s2 - w2)
f(t) = cosh wt , ® F(s) = s / (s2 - w2)
f(t) = e-at sin wt , ® F(s) = w / ((s + a)2 + w2)
f(t) = e-at cos wt , ® F(s) = (s + a) / ((s + a)2 + w2)
f(t) = Jn(w t), ® F(s) = {[(r - s)/w]n}/r, r = Ö(s2 + w2)
F(s) = 1/ (s2 (s + a)), ® f(t) = {1/ a2} {at - 1 - e-at}
F(s) = 1/ ((s + a) (s + b)), ® f(t) = {1/ (b - a)} {e-at - e-bt}
F(s) = s/ ((s + a) (s + b)), ® f(t) = {1/ (b - a)} {be-bt - ae-at}
F(s) = 1/ (s (s + a) (s + b)), ® f(t) = 1/ab [1 + {1/ (a - b)} {be-at - ae-bt}]
F(s) = wn2 / {s2 + 2 z wn s + wn2}, ®
f(t) = {wn /Ö(1 - z2)} { e -z wn t } { sin (wn Ö(1 - z2) t )} For z < 1
F(s) = s / {s2 + 2 z wn s + wn2}, ®
f(t) = { -1 /Ö(1 - z2)} { e -z wn t } { sin (wn Ö(1 - z2) t - f)}
f = tan-1 { Ö(1 - z2) /z} For z < 1
F(s) = wn2 / [s {s2 + 2 z wn s + wn2}], ®
f(t) = {1 - 1 /Ö(1 - z2)} { e -z wn t } { sin (wn Ö(1 - z2) t + f)}
f = tan-1 { Ö(1 - z2) /z} For z < 1
If Laplace Transform of f(t) is F(s), then the following holds:-
L[d/dt (f(t))] = sF(s) - f(0)
L[d2 /dt2 (f(t))] = s2F(s) - sf(0) - f(0)

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Copyright 1996 © Cuthbert A. Nyack.