線形結合のラプラス変換

ラプラス変換

$$\begin{array}{rcl}\mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}f\left(t\right)e^{–st}\mathrm{d}t\end{array}$$

$ag(t)+bh(t)$のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& ag(t)+bh(t)\cdots a,b:t\mathrm{に}\mathrm{よ}\mathrm{ら}\mathrm{な}\mathrm{い}(\mathrm{定}\mathrm{数})\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}\{ag(t)+bh(t)\}{e}^{–st}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}\{ag(t)e^{–st}+bh(t)e^{–st}\}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}ag(t)e^{–st}\mathrm{d}t+{\int }_0^{\mathrm{\infty }}bh(t)e^{–st}\mathrm{d}t\\ & =& a{\int }_0^{\mathrm{\infty }}g(t)e^{–st}\mathrm{d}t+b{\int }_0^{\mathrm{\infty }}h(t)e^{–st}\mathrm{d}t\\ & =& a\mathfrak{L}\left[g\left(t\right)\right]+b\mathfrak{L}\left[h\left(t\right)\right]\end{array}$$