sin凾数のラプラス変換

ラプラス変換

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

$\sin \left(\omega t\right)$のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& \sin \left(\omega t\right)\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}\sin \left(\omega t\right)e^{–st}\mathrm{d}t\\ & =& {[\sin \left(\omega t\right)\cdot \frac{–1}{s}{e}^{-st}]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}\omega \cos \left(\omega t\right)\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,\frac{\mathrm{d}}{\mathrm{d}t}\sin \left(\omega t\right)=\omega \cos \left(\omega t\right)\\ & =& [\sin \left(\omega \mathrm{\infty }\right)\cdot \frac{–1}{s}{e}^{-s\mathrm{\infty }}-\sin \left(\omega 0\right)\cdot \frac{–1}{s}{e}^{-s0}]-\left(\frac{-\omega }{s}\right){\int }_0^{\mathrm{\infty }}\cos \left(\omega t\right)e^{–st}\mathrm{d}t\\ & =& [0-0]+\frac{\omega }{s}{\int }_0^{\mathrm{\infty }}\cos \left(\omega t\right)e^{–st}\mathrm{d}t\\ & & \cdots e^{-\mathrm{\infty }}=0,\sin \left(0\right)=0\\ & =& \frac{\omega }{s}{\int }_0^{\mathrm{\infty }}\cos \left(\omega t\right)e^{–st}\mathrm{d}t\\ & =& \frac{\omega }{s}[{[\cos \left(\omega t\right)\cdot \frac{–1}{s}{e}^{-st}]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}-\omega \sin \left(\omega t\right)\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{微}\mathrm{分})\\ & & \cdots \int e^{at}\mathrm{d}t=\frac{1}{a}{e}^{at}+C,\frac{\mathrm{d}}{\mathrm{d}t}\cos \left(\omega t\right)=-\omega \sin \left(\omega t\right)\\ & =& \frac{\omega }{s}[[\cos \left(\omega \mathrm{\infty }\right)\cdot \frac{–1}{s}{e}^{-s\mathrm{\infty }}-\cos \left(\omega 0\right)\cdot \frac{–1}{s}{e}^{-s0}]-\frac{\omega }{s}{\int }_0^{\mathrm{\infty }}\sin \left(\omega t\right)e^{–st}\mathrm{d}t]\\ & =& \frac{\omega }{s}[[0-1\cdot \frac{–1}{s}\cdot 1]-\frac{\omega }{s}{\int }_0^{\mathrm{\infty }}\sin \left(\omega t\right)e^{–st}\mathrm{d}t]\\ & & \cdots e^{-\mathrm{\infty }}=0,\cos \left(0\right)=1,e^0=1\\ & =& \frac{\omega }{s}[\frac{1}{s}-\frac{\omega }{s}{\int }_0^{\mathrm{\infty }}\sin \left(\omega t\right)e^{–st}\mathrm{d}t]\\ & =& \frac{\omega }{s^2}-\frac{{\omega }^2}{s^2}{\int }_0^{\mathrm{\infty }}\sin \left(\omega t\right)e^{–st}\mathrm{d}t\\ & =& \frac{\omega }{s^2}-\frac{{\omega }^2}{s^2}\mathfrak{L}\left[f\left(t\right)\right]\\ \mathfrak{L}\left[f\left(t\right)\right]+\frac{{\omega }^2}{s^2}\mathfrak{L}\left[f\left(t\right)\right]& =& \frac{\omega }{s^2}\\ \mathfrak{L}\left[f\left(t\right)\right](1+\frac{{\omega }^2}{s^2})& =& \frac{\omega }{s^2}\\ \mathfrak{L}\left[f\left(t\right)\right]& =& \frac{\omega }{s^2}\frac{1}{1+\frac{{\omega }^2}{s^2}}\\ & =& \frac{\omega }{s^2(1+\frac{{\omega }^2}{s^2})}\\ & =& \frac{\omega }{s^2+{\omega }^2}\end{array}$$