cos凾数のラプラス変換

ラプラス変換

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

$\cos \left(\omega t\right)$のラプラス変換

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