eを底とした指数凾数(at乗)とsin凾数との積に対するラプラス変換

ラプラス変換

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

$e^{at}\sin \left(bt\right)$のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& e^{at}\sin \left(bt\right)\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}f\left(t\right)e^{–st}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{e}^{at}\sin \left(bt\right)e^{–st}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{e}^{-st+at}\sin \left(bt\right)\mathrm{d}t\dots e^A{e}^B=e^{A+B}\\ & =& {\int }_0^{\mathrm{\infty }}{e}^{-(s-a)t}\sin \left(bt\right)\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{e}^{-ut}\sin \left(bt\right)\mathrm{d}t\dots u=s-a\\ & =& \mathfrak{L}\left[\sin \left(bt\right)\right]\dots u\mathrm{を}s\mathrm{と}\mathrm{み}\mathrm{な}\mathrm{し}\mathrm{て}\mathrm{利}\mathrm{用}\mathrm{す}\mathrm{る}\\ & =& \frac{b}{u^2+b^2}\dots \mathfrak{L}\left[\sin \left(\omega t\right)\right]＝\frac{\omega }{s^2+{\omega }^2}\\ & =& \frac{b}{{(s-a)}^2+b^2}\end{array}$$