\(f(t)\) \(\Laplace {f(t)}=F(s)\)
(1) \(\;1\) \(\dfrac {1}{s}\)
(2) \(\;e^{at}f(t)\) \(F(s-a)\)
(3) \(\;\mathcal {U}(t-a)\) \(\dfrac {e^{-as}}{s}\)
(4) \(\;f(t-a)\mathcal {U}(t-a)\) \(e^{-as}F(s)\)
(5) \(\;\delta (t)\) \(1\)
(6) \(\;\delta (t-t_0)\) \(e^{-st_0}\)
(7) \(\;t^nf(t)\) \((-1)^n\dfrac {d^nF(s)}{ds^n}\)
(8) \(\;f'(t)\) \(sF(s) - f(0)\)
(9) \(\;f^{n}(t)\) \(s^nF(s) - \displaystyle \sum _{k=0}^{n-1} s^{(n-1-k)} f^{(k)}(0)\)
(10) \(\;\displaystyle {\int _0^t f(x)g(t-x)dx}\) \(F(s)G(s)\)
(11) \(\;t^n\) (\(n=0,1,\dots \)) \(\dfrac {n!}{s^{n+1}}\)
(12) \(\;t^x\) (\(x\geq -1\)) \(\dfrac {\Gamma (x+1)}{s^{x+1}}\)
(13) \(\;\sin (kt)\) \(\dfrac {k}{s^2+k^2}\)
(14) \(\;\cos (kt)\) \(\dfrac {s}{s^2+k^2}\)

(15) \(\;e^{at}\) \(\dfrac {1}{s-a}\)
(16) \(\;\sinh (kt)\) \(\dfrac {k}{s^2-k^2}\)
(17) \(\;\cosh (kt)\) \(\dfrac {s}{s^2-k^2}\)
(18) \(\;\dfrac {e^{at}-e^{bt}}{a-b}\) \(\dfrac {1}{(s-a)(s-b)}\)
(19) \(\;\dfrac {ae^{at}-be^{bt}}{a-b}\) \(\dfrac {s}{(s-a)(s-b)}\)
(20) \(\;te^{at}\) \(\dfrac {1}{(s-a)^2}\)
(21) \(\;t^ne^{at}\) \(\dfrac {n!}{(s-a)^{n+1}}\)
(22) \(\;e^{at}\sin (kt)\) \(\dfrac {k}{(s-a)^2+k^2}\)
(23) \(\;e^{at}\cos (kt)\) \(\dfrac {s-a}{(s-a)^2+k^2}\)
(24) \(\;e^{at}\sinh (kt)\) \(\dfrac {k}{(s-a)^2-k^2}\)
(25) \(\;e^{at}\cosh (kt)\) \(\dfrac {s-a}{(s-a)^2-k^2}\)
(26) \(\;t\sin (kt)\) \(\dfrac {2ks}{(s^2+k^2)^2}\)
(27) \(\;t\cos (kt)\) \(\dfrac {s^2-k^2}{(s^2+k^2)^2}\)
(28) \(\;t\sinh (kt)\) \(\dfrac {2ks}{(s^2-k^2)^2}\)
(29) \(\;t\cosh (kt)\) \(\dfrac {s^2-k^2}{(s^2-k^2)^2}\)
(30) \(\;\dfrac {\sin (at)}{t}\) \(\arctan \dfrac {a}{s}\)