Trasformate di Laplace più significative

Indichiamo con $\lambda$ l'ascissa di convergenza. Inoltre definiamo

$$\hbox{erf}\,(t)={\frac{2}{\sqrt\pi}}\int_0^te^{-x^2}\,dx,\qqu... ...ox{erfc}\,(t)={\frac{2}{\sqrt\pi}}\int_t^{+\infty}e^{-x^2}\,dx.$$

$$F(t),\qquad f(s)={\cal L}(F(t);s)=\int_0^{+\infty}\,e^{-st}f(t)\,dt$$ (1)

$$G(t)=F(c\,t),\ c>0,\qquad g(s)={\frac{1}{c}}f({\frac{s}{c}}),\qquad\lambda_G=c\,\lambda_F$$ (2)

$$G(t)=F(t-t_0),\ t_0>0,\qquad g(s)=e^{-t_0s}\,f(s),\qquad \lambda_G=\lambda_F$$ (3)

$$G(t)=e^{at}F(t),\ a\in\,{\bf C},\qquad g(s)=f(s-a),\qquad \lambda_G=\lambda_F+\hbox{Re}\,a$$ (4)

$$G(t)=-t\,F(t),\qquad g(s)={\frac{d}{ds}}f(s),\qquad \lambda_G=\lambda_F$$ (5)

$$G(t)=F'(t),\qquad g(s)=sf(s)-F(0^+),\qquad\lambda_G=\max\{\lambda_F,\lambda_{F'}\}$$ (6)

$$G(t)=\int_0^tF(\tau)\,d\tau,\qquad g(s)={\frac{f(s)}{s}},\qquad \lambda_G=\max\{0,\lambda_F\}$$ (7)

$$G(t)={\frac{F(t)}{t}},\qquad g(s)=\int_s^\infty f(\tau)\,d\tau,\qquad \lambda_G=\lambda_F$$ (8)

$$F(t)=H(t),\hfill\qquad f(s)={\frac{1}{s}},\qquad\lambda=0$$ (9)

$$F(t)=H(t)e^{\alpha t},\ \alpha\in\,{\bf C},\hfill\qquad f(s)={\frac{1}{s-\alpha}},\qquad\lambda=\hbox{Re}\,\alpha$$ (10)

$$F(t)=H(t)\sin\omega t,\ \omega\in\,{\bf R},\hfill\qquad f(s)={\frac{\omega}{s^2+\omega^2}},\qquad\lambda=0$$ (11)

$$F(t)=H(t)\cos\omega t,\ \omega\in\,{\bf R},\hfill\qquad f(s)={\frac{s}{s^2+\omega^2}},\qquad\lambda=0$$ (12)

$$F(t)=H(t)\sinh\omega t,\ \omega\in\,{\bf R},\hfill\qquad f(s)={\frac{\omega}{s^2-\omega^2}},\qquad\lambda=\vert\omega\vert$$ (13)

$$F(t)=H(t)\cosh\omega t,\ \omega\in\,{\bf R},\hfill\qquad f(s)={\frac{s}{s^2-\omega^2}},\qquad\lambda=\vert\omega\vert$$ (14)

$$F(t)=H(t)\,t^n,\ n\in\,{\bf N},\hfill\qquad f(s)={\frac{n!}{s^{n+1}}},\qquad\lambda=0$$ (15)

$$F(t)=H(t)\,t^\alpha,\ \hbox{Re}\,\alpha>-1,\hfill\qquad f(s)={\frac{\Gamma(\alpha+1)}{s^{\alpha+1}}} ,\qquad\lambda=0$$ (16)

$$F(t)=H(t)\,e^{-t^2},\hfill\qquad f(s)={\frac{\sqrt\pi}{2}}e^{s^2/4}\hbox{erfc}({\frac{s}{2}}) ,\qquad\lambda=-\infty$$ (17)

$$F(t)=H(t)\,\hbox{erf}(t),\hfill\qquad f(s)={\frac{1}{s}}e^{s^2/4}\hbox{erfc}({\frac{s}{2}}),\qquad\lambda=0$$ (18)

$$F(t)=H(t)\,\hbox{erf}(\sqrt t),\hfill\qquad f(s)={\frac{1}{s\sqrt{s+1}}},\qquad\lambda=0$$ (19)

$$F(t)=H(t)\ln t,\hfill\qquad f(s)={\frac{\Gamma'(1)-\log s}{s}},\qquad\lambda=0$$ (20)

$$F(t)=H(t){\frac{1-e^{-t}}{t}},\hfill\qquad f(s)=\ln(1+{\frac{1}{s}}),\qquad\lambda=0$$ (21)

$$F(t)=H(t)J_0(t),\hfill\qquad f(s)={\frac{1}{\sqrt{s^2+1}}},\qquad\lambda=0$$ (22)

$$G(t)=(F_1*F_2)(t),\qquad g(s)=f_1(s)\cdot f_2(s),\qquad \lambda_G=\max\{\lambda_{F_1},\lambda_{F_2}\}$$ (23)

$$G(t)=H(t)F(t):\ F(t+T)=F(t),\ T>0,\hfill\qquad g(s)={\frac{1}{1-e^{-sT}}}\int_0^Te^{-st}F(t)\,dt,\qquad\lambda=0$$ (24)