Tauberian Theorem: If f(t) and its derivative are piecewise continuous functions of exponential order, then \( \displaystyle f(+0) = \lim_{\lambda\to \infty} \,\lambda \int_0^{\infty} f(t)\,e^{-\lambda\,t}\,{\text d}t = \lim_{\lambda\to \infty} \,\lambda \,f^L (\lambda ) . \)

Table of Laplace transforms

It is useful to have a “library” of Laplace transforms at hand; some common ones are listed below.

f(t)	f^L(λ)
\( \displaystyle e^{at}\,f(t) \)	\( \displaystyle f^L (\lambda - a) \)
\( \displaystyle a \,e^{-ab\,t}\, f\left( a\,t \right) \)	\( \displaystyle f^L \left( \frac{\lambda}{a} + b \right) \)
\( \displaystyle e^{at} \)	1/(λ - 𝑎)
sin ωt	\( \displaystyle \frac{\omega}{\lambda^2 + \omega^2} \)
\( \displaystyle t\,\sin \omega t \)	\( \displaystyle \frac{2\omega\lambda}{\left( \lambda^2 + \omega^2 \right)^2} \)
\( \displaystyle e^{kt}\,\sin \omega t \)	\( \displaystyle \frac{\omega}{(\lambda -k )^2 + \omega^2} \)
sinh ωt	\( \displaystyle \frac{\omega}{\lambda^2 - \omega^2} \)
t sinh ωt	\( \displaystyle \frac{2\,\omega\lambda}{\left( \lambda^2 - \omega^2 \right)^2} \)
\( \displaystyle \cos \omega t - \omega t\,\sin \omega t \)	\( \displaystyle \frac{\lambda \left( \lambda^2 - \omega^2 \right)}{\left( \lambda^2 + \omega^2 \right)^2} \)
\( \displaystyle \cos \omega t + \omega t\,\sin \omega t \)	\( \displaystyle \frac{\lambda \left( \lambda^2 + 3\,\omega^2 \right)}{\left( \lambda^2 + \omega^2 \right)^2} \)
t^p	\( \displaystyle \frac{\Gamma (p+1)}{\lambda^{p+1}} \)
\( \displaystyle \left( \pi \,t \right)^{-1/2} \)	\( \displaystyle \lambda^{-1/2} \)
\( \displaystyle \frac{\sin \omega t}{t} \)	\( \displaystyle \arctan \frac{\omega}{\lambda} \)
\( \displaystyle \frac{2}{t} \left( 1 - \cos \omega t \right) \)	\( \displaystyle \ln \left( 1 + \frac{\omega^2}{\lambda^2} \right) \)

f(t)	f^L(λ)
\( \displaystyle f\ast g(t) = \int_0^t f(\tau )\,g(t-\tau )\,{\text d}\tau = g\ast f(t) \)	f^Lg^L
H(t)	1/λ
δ(t)	1
\( \displaystyle t^p\, e^{kt} \)	\( \displaystyle \frac{\Gamma (p+1)}{(\lambda -k)^{p+1}} \)
cos ωt	\( \displaystyle \frac{\lambda}{\lambda^{2} + \omega^2} \)
t cos ωt	\( \displaystyle \frac{\lambda^2 - \omega^2}{\left( \lambda^{2} + \omega^2 \right)^2} \)
\( e^{kt}\cos \omega t \)	\( \displaystyle \frac{\lambda -k}{(\lambda - k)^2 + \omega^2} \)
cosh ωt	\( \frac{\lambda}{\lambda^{2} - \omega^2} \)
t cosh ωt	\( \frac{\lambda^2 + \omega^2}{\left( \lambda^{2} - \omega^2 \right)^2} \)
\( \displaystyle \sin \omega t - \omega t\,\cos \omega t \)	\( \displaystyle \frac{2\,\omega^3}{\left( \lambda^2 + \omega^2 \right)^2} \)
\( \displaystyle \sin \omega t + \omega t\,\cos \omega t \)	\( \displaystyle \frac{2\,\omega \lambda^2}{\left( \lambda^2 + \omega^2 \right)^2} \)
\( \displaystyle \frac{1}{t} \left( 1 - e^{-t} \right) \)	\( \displaystyle \ln \left( 1 + \frac{1}{\lambda} \right) \)
\( \displaystyle \frac{2}{t}\,\sinh at \)	\( \displaystyle \in \frac{\lambda + a}{\lambda - a} \)
\( \displaystyle \frac{2}{t} \left( 1 - \cosh at \right) \)	\( \displaystyle \ln \left( 1 - \frac{a^2}{\lambda^2} \right) \)

Here

\[ f^L = {\cal L} \left[ f(t) \right] (\lambda ) = \int_0^{\infty} e^{-\lambda\,t}f(t)\,{\text d}t , \qquad \Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} {\text d}t , \qquad H(t) = \begin{cases} 1 , & \ \mbox{ if } t > 0, \\ 1/2 , & \ \mbox{ if } t = 0, \\ 0, & \ \mbox{ if } t < 0. \end{cases} \]

Elementary Properties of the Laplace Transforms

Linearity: \( {\cal L} \left[ \alpha\,f(t) + \beta\, g(t) \right] = \alpha\, {\cal L} \left[ f \right] + \beta\,{\cal L} \left[ g \right] = \alpha\, f^L + \beta \, g^L . \)
The derivative rule: \( {\cal L} \left[ f^{(n)} (t) \right] = \lambda^n - \sum_{k=1}^n \lambda^{n-k} f^{(k-1)} (+0) . \)
Convolution rule: \( {\cal L} \left[ f \ast g \right] = f^L \,g^L . \)
Recall that the convolution of two functions f and g is
\[ \left( f \ast g \right) (t) = \int_0^t f(t-\tau )\,g(\tau ) \,{\text d} \tau = \int_0^t g(t-\tau )\,f(\tau ) \,{\text d} \tau = \left( g \ast f\right) (t) . \]
Shift rule: \( {\cal L} \left[ f(t-a)\,H(t-a) \right] = e^{-a\lambda} \,f^L (\lambda ) . \)
Similarity rule: \( {\cal L} \left[ f(kt) \right] = \frac{1}{k}\, f^L \left( \frac{\lambda}{k} \right) . \)
Attenuation rule: \( {\cal L} \left[ e^{-at} \, f(t) \right] = f^L \left( \lambda +a \right) . \)
Differentiation rule: \( \frac{{\text d}}{{\text d} \lambda} \, f^L (\lambda ) = - {\cal L} \left[ t\, f(t) \right] . \)
Integration rule: \( {\cal L} \left[ t^n \ast f(t) \right] = \frac{n!}{\lambda^{n+1}} \, f^L (\lambda ) . \)
The Laplace transform of periodic functions.
If \( f(t) = f(t+ \omega ) , \) then \( \displaystyle f^L (\lambda ) = \int_0^{\infty} f(t )\,e^{-\lambda\,t} \,{\text d} t = \frac{1}{1- e^{-\omega\lambda}} \, \int_0^{\omega} \,f(t ) \,e^{-\lambda \, t} \,{\text d} t . \)
The Laplace transform of anti-periodic functions.
If \( f(t) = -f(t+ \omega ) , \) then \( \displaystyle f^L (\lambda ) = \int_0^{\infty} f(t )\,e^{-\lambda\,t} \,{\text d} t = \frac{1}{1+ e^{-\omega\lambda}} \, \int_0^{\omega} \,f(t ) \,e^{-\lambda \, t} \,{\text d} t . \)

Definition: The full-wave rectifier of a function f(t), defined on a finite interval 0≤t≤T, is a periodic function with period T that is equal to f(t) on the interval [0,T].
The half-wave rectifier of a function f(t), defined on a finite interval 0≤t≤T, is a periodic function with period 2T that coincides with f(t) on the interval [0,T] and is identically zero on the interval [T,2T].

half = Plot[f[t], {t, 0, 4*Pi}, PlotStyle -> Thickness[0.015], AspectRatio -> 1, Axes -> False];
txt = Graphics[Text[Style["f(t)", FontSize -> 14, Red], {1.5, 0.3}]];
t0 = Graphics[Text[Style["0", FontSize -> 14, Black], {-0.1, -0.5}]];
t1 = Graphics[Text[Style["T", FontSize -> 14, Black], {3.1, -0.5}]];
t2 = Graphics[Text[Style["2T", FontSize -> 14, Black], {6.3, -0.5}]];
t3 = Graphics[Text[Style["3T", FontSize -> 14, Black], {9.35, -0.5}]];
txt2 = Graphics[Text[Style["f(t)", FontSize -> 14, Red], {7.8, 0.3}]];
Show[txt, half, t0, t1, t2, t3, txt2]