ラプラス変換_微分

・ 一階微分

\( \Large\mathfrak{ L} \{ \displaystyle \frac{d \ f(t)}{dt} \} =\displaystyle \int_{0}^{ \infty } \displaystyle \frac{d \ f(t)}{dt} \cdot \ e^{-st} dt \)

部分積分から，

\( \Large (f \ g )' = f' \ g + f \ g' \)

\( \Large f' \ g = (f \ g )' - f \ g' \)

\( \Large \int f' \ g = \int (f \ g )' - \int f \ g' \)

\( \Large \int f' \ g = [f \ g ] - \int f \ g' \)

\( \Large f' = \displaystyle \frac{d \ f(t)}{t} f = f(t) \)

\( \Large g =e^{-st} \ g' = -s \ e^{-st} \)

\( \Large \displaystyle \int_{0}^{ \infty } \displaystyle \frac{d \ f(t)}{t} \cdot \ e^{-st} dt
　= \left[ f(t) \cdot e^{-st} \right]_{0}^{\infty} - \int_{0}^{ \infty } \left( - s \ e^{-st} \right) \cdot f(t) \ dt \)

\( \Large \displaystyle　= - f(0) + s \int_{0}^{ \infty } e^{-st} \cdot f(t) \ dt \)

\( \Large \displaystyle　= s \ F(s) - f(0) \)

\( \Large\color{red}{\mathfrak{ L} \{ \displaystyle \frac{d \ f(t)}{t} \} = s \ F(s) - f(0)} \)

・ 二階微分

\( \Large\mathfrak{ L} \{ \displaystyle \frac{d^2 \ f(t)}{dt^2} \} = \mathfrak{ L} \{ \displaystyle \frac{d}{dt} \frac{d \ f(t)}{dt} \}\)

\( \Large = s \ \mathfrak{ L} \{ \displaystyle \frac{d \ f(t)}{dt} \} - \frac{d \ f(0)}{dt}\)

\( \Large = s \ \left\{s \ F(s) - f(0) \right\}- \displaystyle　 \frac{d \ f(0)}{dt}\)

\( \Large \displaystyle = s^2 \ F(s) - s \ f(0) -f'(0)\)

次に，応用として化学反応について考えていきます．