# 微分則

$\mathcal{L}\text{\hspace{0.17em}}\left\{\frac{df\left(t\right)}{dt}\right\}=sF\left(s\right)-f\left(0\right)$

$\mathcal{L}\text{\hspace{0.17em}}\left\{-tf\left(t\right)\right\}=\frac{dF\left(s\right)}{ds}$

$\mathcal{L}\text{\hspace{0.17em}}\left\{{f}^{\left(n\right)}\left(t\right)\right\}={s}^{n}F\left(s\right)$$-\left\{{s}^{n-1}f\left(0\right)+{s}^{n-2}{f}^{\left(1\right)}\left(0\right)+\right\$ $\cdots +s{f}^{\left(n-2\right)}\left(0\right)+{f}^{\left(n-1\right)}\left(0\right)\right\}$

$\mathcal{L}\text{\hspace{0.17em}}\left\{{\left(-t\right)}^{n}f\left(t\right)\right\}={F}^{\left(n\right)}\left(s\right)$

## ■証明

$\mathcal{L}\text{\hspace{0.17em}}\left\{\frac{df\left(t\right)}{dt}\right\}={\int }_{0}^{\infty }{e}^{-st}\frac{df\left(t\right)}{dt}dt$

$={\left[f\left(t\right){e}^{-st}\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\frac{d}{dt}\left\{{e}^{-st}\right\}f\left(t\right)dt$

$={\left[f\left(t\right){e}^{-st}\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\left(-s{e}^{-st}\right)f\left(t\right)dt$

$=-f\left(0\right)+s{\int }_{0}^{\infty }{e}^{-st}f\left(t\right)dt$

$=-f\left(0\right)+sF\left(s\right)$

$=sF\left(s\right)-f\left(0\right)$

## ■証明

$\frac{dF\left(s\right)}{ds}=\frac{d}{ds}\left\{{\int }_{0}^{\infty }{e}^{-st}f\left(t\right)dt\right\}$

$={\int }_{0}^{\infty }\frac{d}{ds}\left({e}^{-st}\right)f\left(t\right)dt$

$={\int }_{0}^{\infty }\left(-t{e}^{-st}\right)f\left(t\right)dt$

$=-{\int }_{0}^{\infty }{e}^{-st}\left\{tf\left(t\right)\right\}dt$

$=-\mathcal{L}\text{\hspace{0.17em}}\left\{tf\left(t\right)\right\}$

よって

$\mathcal{L}\text{\hspace{0.17em}}\left\{-tf\left(t\right)\right\}=\frac{dF\left(s\right)}{ds}$

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