積分のラプラス変換

ラプラス変換

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

${\int }_0^{t}g(u)\mathrm{d}u$のラプラス変換

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