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O método da Transformada de Laplace é uma ferramenta muito poderosa na solução de EDO’s lineares e PVI’s correspondentes.
As principais transformadas de Laplace e suas inversas estão tabelas nas abaixo. Em sua maioria, estas transformadas podem ser obtidas através das propriedades que já foram descritas neste artigo.
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Algumas constantes e funções especiais que são usadas nas tabelas são as seguintes:
a) Função Gamma
\begin{equation}
\Gamma(k)=\int_0^\infty e^{-x}x^{k-1}dx, \qquad (k>0)
\end{equation}
b) Função Bessel modificada de ordem \nu
\begin{equation}
I_\nu(x)=\sum_{m=0}^\infty \frac{1}{m!\Gamma(m+\nu+1)}\left(\frac{x}{2}\right)^{2m+\nu}
\end{equation}
c) Função Bessel de ordem 0
\begin{equation}
J_0(x)=1-\frac{x^2}{2^2(1!)^2}+\frac{x^4}{2^4(2!)^2}-\frac{x^6}{2^6(3!)^2}+\cdots
\end{equation}
d) Integral seno
\begin{equation}
\hbox{Si}\ \!(t)=\int_0^t\frac{\sin(x)}{x}dx
\end{equation}
e) Constante de Euler – Mascheroni
\begin{equation}
\gamma=0.57721566490153286060651209008240243104215933593992 …
\end{equation}
| TABELA DE TRANSFORMADA DE LAPLACE | |
| f(t)=\mathscr{L}^{-1}\{F(s)\} | F(s)=\mathscr{L}\{f(t)\} |
| 1 | \dfrac{1}{s} |
| t | \dfrac{1}{s^2} |
| \dfrac{t^{n-1}}{(n-1)!} | \dfrac{1}{s^n}, \qquad (n=1,2,3,...) |
| \dfrac{1}{\sqrt{\pi t}} | \dfrac{1}{\sqrt{s}} |
| 2\sqrt{\frac{t}{\pi}} | \dfrac{1}{s^{\frac{3}{2}}}, |
| \dfrac{t^{k-1}}{\Gamma(k)} | \dfrac{1}{s^{k}},\qquad (k>0) |
| e^{ at} | \dfrac{1}{s-a} |
| te^{at} | \dfrac{1}{(s-a)^2} |
| \dfrac{1}{(n-1)!}t^{n-1}e^{at} | \dfrac{1}{(s-a)^n},\qquad (n=1,2,3...) |
| \dfrac{1}{\Gamma(k)}t^{k-1}e^{at} | \dfrac{1}{(s-a)^k},\qquad (k>0) |
| \dfrac{1}{a-b}\left(e^{at}-e^{bt}\right) | \dfrac{1}{(s-a)(s-b)},\qquad (a\neq b) |
| \dfrac{1}{a-b}\left(ae^{at}-be^{bt}\right) | \dfrac{s}{(s-a)(s-b)},\qquad (a\neq b) |
| \dfrac{1}{w}\sin(wt) | \dfrac{1}{s^2+w^2} |
| \cos(wt) | \dfrac{s}{s^2+w^2} |
| \dfrac{1}{a}\sinh(at) | \dfrac{1}{s^2-a^2} |
| \cosh(at) | \dfrac{s}{s^2-a^2} |
| \dfrac{1}{w}e^{at}\sin(wt) | \dfrac{1}{(s-a)^2+w^2} |
| e^{at}\cos(wt) | \dfrac{s-a}{(s-a)^2+w^2} |
| \dfrac{1}{w^2}(1-\cos(wt)) | \dfrac{1}{s(s^2+w^2)} |
| \dfrac{1}{w^3}(wt-\sin(wt)) | \dfrac{1}{s^2(s^2+w^2)} |
| \dfrac{1}{2w^3}(\sin(wt)-wt\cos(wt)) | \dfrac{1}{(s^2+w^2)^2} |
| \dfrac{t}{2w}\sin(wt) | \dfrac{s}{(s^2+w^2)^2} |
| \dfrac{1}{2w}(\sin(wt)+wt\cos(wt)) | \dfrac{s^2}{(s^2+w^2)^2} |
| \dfrac{1}{b^2-a^2}(\cos(at)-\cos(bt)) | \dfrac{s}{(s^2+a^2)(s^2+b^2)},\qquad (a^2\neq b^2) |
| \dfrac{1}{4a^3}(\sin(at)\cosh(at)-\cos(at)\sinh(at) | \dfrac{1}{(s^4+4a^4)} |
| \dfrac{1}{2a^2}\sin(at)\sinh(at)) | \dfrac{s}{(s^4+4a^4)} |
| \dfrac{1}{2a^3}(\sinh(at)-\sin(at)) | \dfrac{1}{(s^4-a^4)} |
| \dfrac{1}{2a^2}(\cosh(at)-\cos(at)) | \dfrac{s}{(s^4-a^4)} |
| \dfrac{1}{2a^2}(\cosh(at)-\cos(at)) | \dfrac{s}{(s^4-a^4)} |
| \dfrac{1}{2\sqrt{\pi t^3}}(e^{bt}-e^{at} | \sqrt{s-a}-\sqrt{s-b} |
| e^{\frac{-(a+b)t}{2}}I_0\left(\frac{a-b}{2}t\right) | \dfrac{1}{\sqrt{s+a}\sqrt{s+b}} |
| J_0(at) | \dfrac{1}{\sqrt{s^2+a^2}} |
| \dfrac{1}{\sqrt{\pi t}}e^{at}(1+2at) | \dfrac{s}{(s-a)^{\frac{3}{2}}} |
| \dfrac{\sqrt{\pi}}{\Gamma(k)}\left(\frac{t}{2a}\right)^{k-\frac{1}{2}}I_{k-\frac{1}{2}}(at) | \dfrac{1}{(s^2-a^2)^k},\qquad (k>0) |
| J_0(2\sqrt{kt}) | \dfrac{1}{s}e^{-\frac{k}{s}},\qquad (k>0) |
| \dfrac{1}{\sqrt{\pi t}}\cos(2\sqrt{k t}) | \dfrac{1}{\sqrt{s}}e^{-\frac{k}{s}} |
| \dfrac{1}{\sqrt{\pi t}}\sinh(2\sqrt{k t}) | \dfrac{1}{s^{\frac{3}{2}}}e^{\frac{k}{s}} |
| \dfrac{k}{2\sqrt{\pi t^3}}e^{-\frac{k^2}{4t}} | e^{-k\sqrt{s}},\qquad (k>0) |
| -\ln(t)-\gamma,\qquad (\gamma\approx 0,5772) | \dfrac{1}{s}\ln(s) |
| \dfrac{1}{t}\left(e^{bt}-e^{at}\right) | \ln\left(\dfrac{s-a}{s-b}\right) |
| \dfrac{2}{t}\left(1-\cos(wt)\right) | \ln\left(\dfrac{s^2+w^2}{s^2}\right) |
| \dfrac{2}{t}\left(1-\cosh(at)\right) | \ln\left(\frac{s^2-a^2}{s^2}\right) |
| \dfrac{1}{t}\sin(wt) | \tan^{-1}\left(\frac{w}{s}\right) |
| \hbox{Si}\ \!(t) | \frac{1}{s}\cot^{-1}(s) |
| \alpha f(t)+\beta g(t) | \alpha\mathscr{L}{L}\left\{ f(t)\right\}+\beta\mathscr{L}{L}\left\{g(t)\right\} |
| f'(t) | s\mathscr{L}{L}\left\{f(t)\right\}-f(0) |
| f''(t) | s^2\mathscr{L}{L}\left\{f(t)\right\}-sf(0)-f'(0) |
| e^{at}f(t) | F(s-a) |
| u(t-a)f(t-a) | e^{-as}F(s) |
| u(t-a) | \dfrac{e^{-as}}{s} |
| \int_0^t f(\tau)d\tau | \dfrac{F(s)}{s} |
| \delta(t-a) | e^{-as} |
| (f*g)(t) | F(s)G(s) |
| tf(t) | -\dfrac{dF(s)}{ds} |
| \dfrac{f(t)}{t} | \int_s^\infty F(\hat{s})d\hat{s} |
| \dfrac{1}{\sqrt{\pi t^3}} e^{-a^2/4t} | \dfrac{e^{-a \sqrt{s}}}{\sqrt{s}} |
| \dfrac{a}{2\sqrt{\pi t^3}} e^{-a^2/4t} | e^{-a \sqrt{s}} |
| erfc \left( \dfrac{a}{2 \sqrt{t}} \right) | \dfrac{e^{-a \sqrt{s}}}{s} |
| 2 \sqrt{\dfrac{t}{\pi} }e^{-a^2/4t} - a erfc \left( \dfrac{a}{2 \sqrt{t}} \right) | \dfrac{e^{-a \sqrt{s}}}{s \sqrt{s}} |
| e^{ab} e^{b^2 t} a erfc \left( b \sqrt{t} + \dfrac{a}{2 \sqrt{t}} \right) | \dfrac{e^{-a \sqrt{s}}}{\sqrt{s} ( \sqrt{s} + b)} |
| e^{ab} e^{b^2 t} a erfc \left( b \sqrt{t} + \dfrac{a}{2 \sqrt{t}} \right) + erfc \left( \dfrac{a}{2 \sqrt{t}} \right) | \dfrac{ b e^{-a \sqrt{s}}}{\sqrt{s} ( \sqrt{s} + b)} |
| erf(\sqrt{t}) | \dfrac{1}{s \sqrt{s+1}} |
| erfc(\sqrt{t}) | \dfrac{1}{s} \left[ 1 - \dfrac{1}{\sqrt{s+1}} \right] |
| e^t erfc(\sqrt{t}) | \dfrac{1}{\sqrt{s} (s+1)} |
| e^t erf(\sqrt{t}) | \dfrac{1}{\sqrt{s} (s-1)} |
| e^{-Gt/C} erf \left( \dfrac{x}{2} \sqrt{\dfrac{RC}{t}} \right) | \dfrac{C}{Cs + G} \left( 1 - e^{-x \sqrt{RC s + RG}} \right) sendo C, G, R e x constantes |
| \sum \limits_{n=0}^{\infty}{\left[ erf\left( \dfrac{2n +1 +a}{2 \sqrt{t}} \right) - erf\left( \dfrac{2n +1 - a}{2 \sqrt{t}} \right) \right]} | \dfrac{\sinh a \sqrt{s} }{s \sinh \sqrt{s} } |