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Transformada de Laplace - Um Livro Colaborativo

A.1 Tabelas de Transformadas de Laplace

As principais transformadas de Laplace e suas inversas estão tabelas nas tabelas A.1 e A.2. Algumas constantes e funções especiais que são usadas nas tabelas são as seguintes:

a)
Função Gamma
 $\Gamma \left(k\right)={\int }_{0}^{\infty }{e}^{-x}{x}^{k-1}dx,\phantom{\rule{2em}{0ex}}\left(k>0\right)$ (A.1)
b)
Função Bessel modiﬁcada de ordem $\nu$
 ${I}_{\nu }\left(x\right)=\sum _{m=0}^{\infty }\frac{1}{m!\Gamma \left(m+\nu +1\right)}{\left(\frac{x}{2}\right)}^{2m+\nu }$ (A.2)
c)
Função Bessel de ordem $0$
 ${J}_{0}\left(x\right)=1-\frac{{x}^{2}}{{2}^{2}{\left(1!\right)}^{2}}+\frac{{x}^{4}}{{2}^{4}{\left(2!\right)}^{2}}-\frac{{x}^{6}}{{2}^{6}{\left(3!\right)}^{2}}+\cdots$ (A.3)
d)
Integral seno
 $Si\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left(t\right)={\int }_{0}^{t}\frac{sen\phantom{\rule{0.3em}{0ex}}\left(x\right)}{x}dx$ (A.4)
e)
Constante de Euler - Mascheroni
 $\gamma =0.57721566490153286060651209008240243104215933593992...$ (A.5)

 $F\left(s\right)=\mathsc{L}\left\{f\left(t\right)\right\}$ $f\left(t\right)={\mathsc{L}}^{-1}\left\{F\left(s\right)\right\}$ $\frac{1}{s}$ $1$ $\frac{1}{{s}^{2}}$ $t$ $\frac{1}{{s}^{n}},\phantom{\rule{2em}{0ex}}\left(n=1,2,3,...\right)$ $\frac{{t}^{n-1}}{\left(n-1\right)!}$ $\frac{1}{\sqrt{s}},$ $\frac{1}{\sqrt{\pi t}}$ $\frac{1}{{s}^{\frac{3}{2}}},$ $2\sqrt{\frac{t}{\pi }}$ $\frac{1}{{s}^{k}},\phantom{\rule{2em}{0ex}}\left(k>0\right)$ $\frac{{t}^{k-1}}{\Gamma \left(k\right)}$ $\frac{1}{s-a}$ ${e}^{at}$ $\frac{1}{{\left(s-a\right)}^{2}}$ $t{e}^{at}$ $\frac{1}{{\left(s-a\right)}^{n}},\phantom{\rule{2em}{0ex}}\left(n=1,2,3...\right)$ $\frac{1}{\left(n-1\right)!}{t}^{n-1}{e}^{at}$ $\frac{1}{{\left(s-a\right)}^{k}},\phantom{\rule{2em}{0ex}}\left(k>0\right)$ $\frac{1}{\Gamma \left(k\right)}{t}^{k-1}{e}^{at}$ $\frac{1}{\left(s-a\right)\left(s-b\right)},\phantom{\rule{2em}{0ex}}\left(a\ne b\right)$ $\frac{1}{a-b}\left({e}^{at}-{e}^{bt}\right)$ $\frac{s}{\left(s-a\right)\left(s-b\right)},\phantom{\rule{2em}{0ex}}\left(a\ne b\right)$ $\frac{1}{a-b}\left(a{e}^{at}-b{e}^{bt}\right)$ $\frac{1}{{s}^{2}+{w}^{2}}$ $\frac{1}{w}sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)$ $\frac{s}{{s}^{2}+{w}^{2}}$ $cos\left(wt\right)$ $\frac{1}{{s}^{2}-{a}^{2}}$ $\frac{1}{a}senh\phantom{\rule{0.3em}{0ex}}\left(at\right)$ $\frac{s}{{s}^{2}-{a}^{2}}$ $cosh\left(at\right)$ $\frac{1}{{\left(s-a\right)}^{2}+{w}^{2}}$ $\frac{1}{w}{e}^{at}sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)$ $\frac{s-a}{{\left(s-a\right)}^{2}+{w}^{2}}$ ${e}^{at}cos\left(wt\right)$ $\frac{1}{s\left({s}^{2}+{w}^{2}\right)}$ $\frac{1}{{w}^{2}}\left(1-cos\left(wt\right)\right)$ $\frac{1}{{s}^{2}\left({s}^{2}+{w}^{2}\right)}$ $\frac{1}{{w}^{3}}\left(wt-sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)\right)$ $\frac{1}{{\left({s}^{2}+{w}^{2}\right)}^{2}}$ $\frac{1}{2{w}^{3}}\left(sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)-wtcos\left(wt\right)\right)$ $\frac{s}{{\left({s}^{2}+{w}^{2}\right)}^{2}}$ $\frac{t}{2w}sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)$

Tabela A.1: Tabela de transformadas de Laplace - parte 1

 $F\left(s\right)=\mathsc{L}\left\{f\left(t\right)\right\}$ $f\left(t\right)={\mathsc{L}}^{-1}\left\{F\left(s\right)\right\}$ $\frac{{s}^{2}}{{\left({s}^{2}+{w}^{2}\right)}^{2}}$ $\frac{1}{2w}\left(sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)+wtcos\left(wt\right)\right)$ $\frac{s}{\left({s}^{2}+{a}^{2}\right)\left({s}^{2}+{b}^{2}\right)},\phantom{\rule{2em}{0ex}}\left({a}^{2}\ne {b}^{2}\right)$ $\frac{1}{{b}^{2}-{a}^{2}}\left(cos\left(at\right)-cos\left(bt\right)\right)$ $\frac{1}{\left({s}^{4}+4{a}^{4}\right)}$ $\frac{1}{4{a}^{3}}\left(sen\phantom{\rule{0.3em}{0ex}}\left(at\right)cosh\left(at\right)-cos\left(at\right)senh\phantom{\rule{0.3em}{0ex}}\left(at\right)\right)$ $\frac{s}{\left({s}^{4}+4{a}^{4}\right)}$ $\frac{1}{2{a}^{2}}sen\phantom{\rule{0.3em}{0ex}}\left(at\right)senh\phantom{\rule{0.3em}{0ex}}\left(at\right)\right)$ $\frac{1}{\left({s}^{4}-{a}^{2}\right)}$ $\frac{1}{2{a}^{3}}\left(senh\phantom{\rule{0.3em}{0ex}}\left(at\right)-sen\phantom{\rule{0.3em}{0ex}}\left(at\right)\right)$ $\frac{s}{\left({s}^{4}-{a}^{4}\right)}$ $\frac{1}{2{a}^{2}}\left(cosh\left(at\right)-cos\left(at\right)\right)$ $\sqrt{s-a}-\sqrt{s-b}$ $\frac{1}{2\sqrt{\pi {t}^{3}}}\left({e}^{bt}-{e}^{at}\right)$ $\frac{1}{\sqrt{s+a}\sqrt{s+b}}$ ${e}^{\frac{-\left(a+b\right)t}{2}}{I}_{0}\left(\frac{a-b}{2}t\right)$ $\frac{1}{\sqrt{{s}^{2}+{a}^{2}}}$ ${J}_{0}\left(at\right)$ $\frac{s}{{\left(s-a\right)}^{\frac{3}{2}}}$ $\frac{1}{\sqrt{\pi t}}{e}^{at}\left(1+2at\right)$ $\frac{1}{{\left({s}^{2}-{a}^{2}\right)}^{k}},\phantom{\rule{2em}{0ex}}\left(k>0\right)$ $\frac{\sqrt{\pi }}{\Gamma \left(k\right)}{\left(\frac{t}{2a}\right)}^{k-\frac{1}{2}}{I}_{k-\frac{1}{2}}\left(at\right)$ $\frac{1}{s}{e}^{-\frac{k}{s}},\phantom{\rule{2em}{0ex}}\left(k>0\right)$ ${J}_{0}\left(2\sqrt{kt}\right)$ $\frac{1}{\sqrt{s}}{e}^{-\frac{k}{s}}$ $\frac{1}{\sqrt{\pi t}}cos\left(2\sqrt{kt}\right)$ $\frac{1}{{s}^{\frac{3}{2}}}{e}^{\frac{k}{s}}$ $\frac{1}{\sqrt{\pi t}}senh\phantom{\rule{0.3em}{0ex}}\left(2\sqrt{kt}\right)$ ${e}^{-k\sqrt{s}},\phantom{\rule{2em}{0ex}}\left(k>0\right)$ $\frac{k}{2\sqrt{\pi {t}^{3}}}{e}^{-\frac{{k}^{2}}{4t}}$ $\frac{1}{s}ln\left(s\right)$ $-ln\left(t\right)-\gamma ,\phantom{\rule{2em}{0ex}}\left(\gamma \approx 0,5772\right)$ $ln\left(\frac{s-a}{s-b}\right)$ $\frac{1}{t}\left({e}^{bt}-{e}^{at}\right)$ $ln\left(\frac{{s}^{2}+{w}^{2}}{{s}^{2}}\right)$ $\frac{2}{t}\left(1-cos\left(wt\right)\right)$ $ln\left(\frac{{s}^{2}-{a}^{2}}{{s}^{2}}\right)$ $\frac{2}{t}\left(1-cosh\left(at\right)\right)$ ${tan}^{-1}\left(\frac{w}{s}\right)$ $\frac{1}{t}sen\phantom{\rule{0.3em}{0ex}}\left(wt\right)$ $\frac{1}{s}{cot}^{-1}\left(s\right)$ $Si\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left(t\right)$

Tabela A.2: Tabela de transformadas de Laplace - parte 2