積分則

$3$\displaystyle{L\text{\hspace{1}} \{ \int _{0}^{t} f $t$ dt \} =\frac{\hspace{2} F $s$ \hspace{2}}{\hspace{2}s\hspace{2}}+\frac{\hspace{2} {f}^{ $ - 1 $ } $0$ \hspace{2}}{\hspace{2}s\hspace{2}}}$

ただし， $3$\displaystyle{{f}^{ $ - 1 $ } $0$ =\int _{ - \infty }^{0} f $t$ dt}$

$3$\displaystyle{L\text{\hspace{1}} \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ \} =\int _{s}^{\infty} F $s$ ds}$

一般に

$3$\displaystyle{ L \{ \int _{0}^{t} \int _{0}^{t} \int _{0}^{t} \cdots \int _{0}^{t} f $t$ { $ dt $ }^{n} \} }$ $3$\displaystyle{ =\frac{\hspace{2} F $s$ \hspace{2}}{\hspace{2}{s}^{n}\hspace{2}}+\frac{\hspace{2} {f}^{ $ - 1 $ } $0$ \hspace{2}}{\hspace{2}{s}^{n}\hspace{2}}+\frac{\hspace{2} {f}^{ $ - 2 $ } $0$ \hspace{2}}{\hspace{2}{s}^{ n- 1 }\hspace{2}}+ }$ $3$\displaystyle{ \cdots +\frac{\hspace{2} {f}^{ $ - n $ } $0$ \hspace{2}}{\hspace{2}s\hspace{2}} }$

ただし

$3$\displaystyle{\displaystyle{{f}^{ $ - n $ } $0$ =\int _{ - \text{\infty} }^{0} \int _{ - \text{\infty} }^{0} \int _{ - \text{\infty} }^{0} }}$ $3$\displaystyle{ \cdots \int _{ - \text{\infty} }^{0} f $t${ $ dt $ }^{n} }$

$3$\displaystyle{L \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{t}^{n}\hspace{2}}f $t$ \} =\int _{s}^{ \text{\infty} } \int _{s}^{ \text{\infty} } \int _{s}^{ \text{\infty} }}$ $3$\displaystyle{ \cdots \int _{s}^{ \text{\infty} }{ F $s$ $ ds $ }^{n} }$

■証明

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

$3$\displaystyle{=\displaystyle{ \int _{0}^{\infty} { $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{e}^{ - st } $ }^{\prime } \{ \displaystyle{ \int _{0}^{t} f $t$ dt } \} dt }}$

$3$\displaystyle{ = \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{e}^{ - st }\displaystyle{\int _{0}^{t} f $t$ dt } \] _{0}^{ \text{\infty} } }$ $3$\displaystyle{ - \displaystyle{\int _{0}^{ \text{\infty} } $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{e}^{ - st } $ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}} \{\displaystyle{\int _{0}^{t} f $t$ dt }\} dt } }$

$3$\displaystyle{ = \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{e}^{ - st }\displaystyle{\int _{0}^{t} f $t$ dt } \] _{0}^{ \text{\infty} } }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}\displaystyle{\int _{0}^{ \text{\infty} } {e}^{ - st }f $t$ dt } }$

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

$3$\displaystyle{=\frac{\hspace{2} F $s$ \hspace{2}}{\hspace{2}s\hspace{2}}}$

■証明

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

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

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

$3$\displaystyle{=\displaystyle{ \int _{0}^{\infty} f $t$ \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}{e}^{ - st } \] _{s}^{\infty}dt }}$

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

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

$3$\displaystyle{=L\text{\hspace{1}} \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ \} }$

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　最終更新日： 2023年6月6日