微分則

$3$\displaystyle{L\text{\hspace{1} } \{ \frac{\hspace{2} df $t$ \hspace{2}}{\hspace{2} dt \hspace{2}} \} =sF $s$ - f $0$ }$

$3$\displaystyle{L\text{\hspace{1} } \{ - tf $t$ \} =\frac{\hspace{2} dF $s$ \hspace{2}}{\hspace{2} ds \hspace{2}}}$

ただし， $3$\displaystyle{L\text{\hspace{1}}\left{{ f\left({t}\right) }\right}=F\left({s}\right)}$

整数 $3$\displaystyle{n\geq 1}$ のとき

$3$\displaystyle{ L\text{\hspace{1} } \{ {f}^{ $n$ } $t$ \} ={s}^{n}F $s$ }$ $3$\displaystyle{ - \left{{ {s}^{ n- 1 }f $0$ +{s}^{ n- 2 }{f}^{ $1$ } $0$ + }$ $3$\displaystyle{ \cdots +s{f}^{ $ n- 2 $ } $0$ +{f}^{ $ n- 1 $ } $0$ \} }$

$3$\displaystyle{L\text{\hspace{1} } \{ { $ - t $ }^{n}f $t$ \} ={F}^{ $n$ } $s$ }$

■証明

$3$\displaystyle{L\text{\hspace{1} } \{ \frac{\hspace{2} df $t$ \hspace{2}}{\hspace{2} dt \hspace{2}} \} =\displaystyle{ \int _{0}^{\infty} {e}^{ - st } }\frac{\hspace{2} df $t$ \hspace{2}}{\hspace{2} dt \hspace{2}}dt}$

部分積分を用いると

$3$\displaystyle{= \[ f $t$ {e}^{ - st } \] _{0}^{\infty}}$ $3$\displaystyle{- \displaystyle{ \int _{0}^{\infty} \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}} \{ {e}^{ - st } \} f $t$ dt }}$

$3$\displaystyle{= \[ f $t$ {e}^{ - st } \] _{0}^{\infty}}$ $3$\displaystyle{- \displaystyle{ \int _{0}^{\infty} $ - s{e}^{ - st } $ f $t$ dt }}$

$3$\displaystyle{=- f $0$ +s\displaystyle{ \int _{0}^{\infty} {e}^{ - st }f $t$ dt }}$

$3$\displaystyle{=- f $0$ +sF $s$ }$

$3$\displaystyle{=sF $s$ - f $0$ }$

■証明

$3$\displaystyle{\frac{\hspace{2} dF $s$ \hspace{2}}{\hspace{2} ds \hspace{2}}=\frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}} \{ \displaystyle{ \int _{0}^{\infty} {e}^{ - st }f $t$ dt } \} }$

$3$\displaystyle{=\displaystyle{ \int _{0}^{\infty} \frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}} $ {e}^{ - st } $ }f $t$ dt}$

$3$\displaystyle{=\displaystyle{ \int _{0}^{\infty} $ - t{e}^{ - st } $ }f $t$ dt}$

$3$\displaystyle{=- \displaystyle{ \int _{0}^{\infty} {e}^{ - st } } \{ tf $t$ \} dt}$

$3$\displaystyle{=- L\text{\hspace{1} } \{ tf $t$ \} }$

よって

$3$\displaystyle{L\text{\hspace{1} } \{ - tf $t$ \} =\frac{\hspace{2} dF $s$ \hspace{2}}{\hspace{2} ds \hspace{2}}}$

●整数 $3$\displaystyle{n\geq 1}$ のとき

$3$\displaystyle{L\text{\hspace{1}}\left{{ {f}^{ \left({n}\right) }\left({t}\right) }\right}}$ $3$\displaystyle{=L\text{\hspace{1}}\left{{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}}{f}^{ \left({ n- 1 }\right) }\left({t}\right) }\right}}$

$3$\displaystyle{=sL\text{\hspace{1}}\left{{ {f}^{ \left({ n- 1 }\right) }\left({t}\right) }\right}}$ $3$\displaystyle{- {f}^{ \left({ n- 1 }\right) }\left({0}\right)}$

$3$\displaystyle{=sL\text{\hspace{1}}\left{{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}}{f}^{ \left({ n- 2 }\right) }\left({t}\right) }\right}}$ $3$\displaystyle{- {f}^{ \left({ n- 1 }\right) }\left({0}\right)}$

$3$\displaystyle{=s\left[{ sL\text{\hspace{1}}\left{{ {f}^{ \left({ n- 2 }\right) }\left({t}\right) }\right}- {f}^{ \left({ n- 2 }\right) }\left({0}\right) }\right]}$ $3$\displaystyle{- {f}^{ \left({ n- 1 }\right) }\left({0}\right)}$

$3$\displaystyle{={s}^{2}L\text{\hspace{1}}\left{{ {f}^{ \left({ n- 2 }\right) }\left({t}\right) }\right}}$ $3$\displaystyle{- \left{{ s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{={s}^{2}L\text{\hspace{1}}\left{{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}}{f}^{ \left({ n- 3 }\right) }\left({t}\right) }\right}}$ $3$\displaystyle{- \left{{ s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{={s}^{2}\left[{ sL\left{{ {f}^{ \left({ n- 3 }\right) }\left({t}\right) }\right}- {f}^{ \left({ n- 3 }\right) }\left({0}\right) }\right]}$ $3$\displaystyle{- \left{{ s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{={s}^{3}L\left{{ {f}^{ \left({ n- 3 }\right) }\left({t}\right) }\right}}$ $3$\displaystyle{- \left{{ {s}^{2}{f}^{ \left({ n- 3 }\right) }\left({0}\right)+s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{\cdots }$

$3$\displaystyle{={s}^{n}L\left{{ f\left({t}\right) }\right}}$ $3$\displaystyle{- \left{{ {s}^{ n- 1 }f\left({0}\right)+{s}^{ n- 2 }{f}^{ \left({1}\right) }\left({0}\right)+\cdots +s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{={s}^{n}F\left({s}\right)}$ $3$\displaystyle{- \left{{ {s}^{ n- 1 }f\left({0}\right)+{s}^{ n- 2 }{f}^{ \left({1}\right) }\left({0}\right)+\cdots +s{f}^{ \left({ n- 2 }\right) }\left({0}\right)+{f}^{ \left({ n- 1 }\right) }\left({0}\right) }\right}}$

$3$\displaystyle{L\left{{ { \left({ - t }\right) }^{n}f\left({t}\right) }\right}}$ $3$\displaystyle{=L\left{{ \left({ - t }\right)\left{{ { \left({ - t }\right) }^{ n- 1 }f\left({t}\right) }\right} }\right}}$

$3$\displaystyle{=\frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}}L\left{{ { \left({ - t }\right) }^{ n- 1 }f\left({t}\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}}\left[{ L\text{\hspace{1}}\left{{ \left({ - t }\right){ \left({ - t }\right) }^{ n- 2 }f\left({t}\right) }\right} }\right]}$

$3$\displaystyle{=\frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}}\left[{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} ds \hspace{2}}L\left{{ { \left({ - t }\right) }^{ n- 2 }f\left({t}\right) }\right}\text{\hspace{1}} }\right]}$

$3$\displaystyle{=\frac{\hspace{2} {d}^{2} \hspace{2}}{\hspace{2} d{s}^{2} \hspace{2}}L\text{\hspace{1}}\left{{ { \left({ - t }\right) }^{ n- 2 }f\left({t}\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2} {d}^{3} \hspace{2}}{\hspace{2} d{s}^{3} \hspace{2}}L\text{\hspace{1}}\left{{ { \left({ - t }\right) }^{ n- 3 }f\left({t}\right) }\right}}$

$3$\displaystyle{\cdots }$

$3$\displaystyle{=\frac{\hspace{2} {d}^{ n- 1 } \hspace{2}}{\hspace{2} d{s}^{ n- 1 } \hspace{2}}L\text{\hspace{1}}\left{{ - tf\left({t}\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2} {d}^{n} \hspace{2}}{\hspace{2} d{s}^{n} \hspace{2}}L\text{\hspace{1}}\left{{ f\left({t}\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2} {d}^{n} \hspace{2}}{\hspace{2} d{s}^{n} \hspace{2}}F\left({s}\right)}$

$3$\displaystyle{={F}^{ \left({n}\right) }\left({s}\right)}$

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最終更新日： 2024年8月24日

微分則

■証明

■証明

●整数 のとき

●整数 $3$\displaystyle{n\geq 1}$ のとき