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Section 5 Mathematical Formulas5-3-1 What's Laplace Transform? The Laplace transform is what the function, ''$f (t)$'' of the time, ''$t$'' is converted to the function, ''$F(s)$'' of the complex number, ''$s$'', and is indicated in the following equation. \begin{eqnarray} F(s)=\mathcal{L}[f(t)] =\int_0^\infty f(t)e^{-st}\;dt \nonumber \end{eqnarray}In addition, the Laplace inverse transform is what the function, ''$F(s)$'' of the complex number, ''$s$'' is returned to the function, ''$f (t)$'' of the time, ''$t$'', and is indicated in the following equation. \begin{eqnarray} f(t)=\mathcal{L}^{-1}[F(s)]=\lim_{p \to \infty}\frac{1}{2\pi\;i}\int_{c-ip}^{c+ip} F(s)e^{st}\;ds \nonumber \end{eqnarray}Note that it would be convenient to utilize the Table 5-3-2 ''Laplace transform table'', when the Laplace transform or its inverse transform of the major functions is implemented, as it would not be practical to perform the integration above in every case. |