単位ステップ凾数のラプラス変換

ラプラス変換

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

単位ステップ凾数

$$\begin{array}{rcl}u(t)& =& \{\begin{array}{c}1(t\ge 0)\\ 0(t<0)\end{array}\end{array}$$

単位ステップ凾数のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& u(t)\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}u(t)e^{–st}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}1e^{–st}\mathrm{d}t\dots \mathrm{ス}\mathrm{テ}\mathrm{ッ}\mathrm{プ}\mathrm{入}\mathrm{力}\mathrm{の}\mathrm{定}\mathrm{義}\mathrm{よ}\mathrm{り}\mathrm{積}\mathrm{分}\mathrm{区}\mathrm{間}\mathrm{で}\mathrm{は}1\\ & =& {\int }_0^{\mathrm{\infty }}\frac{–1}{s}{e}^a\mathrm{d}a\dots a=-st,\frac{\mathrm{d}a}{\mathrm{d}t}=-s,\mathrm{d}t=\frac{-1}{s}\mathrm{d}a\\ & =& \frac{–1}{s}{\int }_0^{\mathrm{\infty }}{e}^a\mathrm{d}a\\ & =& \frac{–1}{s}{\left[e^{-st}\right]}_0^{\mathrm{\infty }}\dots {\int }_0^{\mathrm{\infty }}{e}^a\mathrm{d}a=e^a+C(\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数})\\ & =& \frac{–1}{s}[e^{-\mathrm{\infty }t}-e^{-0t}]\\ & =& \frac{–1}{s}[0-1]\dots e^{-\mathrm{\infty }}=0,e^0=1\\ & =& \frac{–1}{s}\cdot -1\\ & =& \frac{1}{s}\end{array}$$