微分のラプラス変換

ラプラス変換

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

$\frac{\mathrm{d}f}{\mathrm{d}t}$のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\\ \mathfrak{L}\left[g\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & =& {\left[g\left(t\right)e^{-st}\right]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\left(–s\right)e^{–st}\mathrm{d}t\\ & & \cdots {\int }_a^b{f}^{\prime }\left(t\right)g\left(t\right)\mathrm{d}t={\left[f\left(t\right)g\left(t\right)\right]}_a^b-{\int }_a^{b}f\left(t\right)g^{\prime }\left(t\right)\mathrm{d}t(f^{\prime }:f\mathrm{の}\mathrm{微}\mathrm{分},g^{\prime }:g\mathrm{の}\mathrm{部}\mathrm{分})\\ & & \cdots \frac{\mathrm{d}}{\mathrm{d}t}{e}^{at}=a{e}^{at}\\ & =& [g\left(\mathrm{\infty }\right)e^{-s\mathrm{\infty }}-g\left(0\right)e^{-s0}]-\left(–s\right){\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & =& [0-g\left(0\right)\cdot 1]-\left(–s\right){\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & & \cdots e^{-\mathrm{\infty }}=0,e^0=1\\ & =& -g\left(0\right)+s{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & =& -g\left(0\right)+s\mathfrak{L}\left[g\left(t\right)\right]\\ & =& s\mathfrak{L}\left[g\left(t\right)\right]-g\left(0\right)\end{array}$$