ラプラス変換基本公式表

詳細は各公式のページを参照してください．

基本公式	$3$\displaystyle{f $t$ }$ (実数 $3$\displaystyle{t}$ の関数)	$3$\displaystyle{F $s$ }$ ( $3$\displaystyle{s}$ の関数)
線形性	$3$\displaystyle{ {\sum }\limits_{n} \hspace{1}{a}_{n} f $t$ }$	$3$\displaystyle{ {\sum }\limits_{n} \hspace{1}{a}_{n} \hspace{1}{F}_{n} $s$ }$
相似定理	$3$\displaystyle{f $ at $ }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}F $ \frac{\hspace{2}s\hspace{2}}{\hspace{2}a\hspace{2}} $ }$
相似定理	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}f $ \frac{\hspace{2}t\hspace{2}}{\hspace{2}a\hspace{2}} $ }$	$3$\displaystyle{F $ as $ }$
推移則	$3$\displaystyle{f $ t- a $ u $ t- a $ }$	$3$\displaystyle{{e}^{ - as }F $s$ }$
推移則	$3$\displaystyle{{e}^{ - at }f $t$ }$	$3$\displaystyle{F $ s+a $ }$
微分則	$3$\displaystyle{ \frac{\hspace{2} df $t$ \hspace{2}}{\hspace{2} dt \hspace{2}} }$	$3$\displaystyle{ sF $s$ - f $0$ }$
	$3$\displaystyle{ \frac{\hspace{2} {d}^{2}f $t$ \hspace{2}}{\hspace{2} d{t}^{2} \hspace{2}} }$	$3$\displaystyle{ {s}^{2}F $s$ - sf $0$ - {f}^{\prime } $0$ }$
	$3$\displaystyle{ \frac{\hspace{2} {d}^{n}f $t$ \hspace{2}}{\hspace{2} d{t}^{n} \hspace{2}} }$	$3$\displaystyle{ {s}^{n}F $s$ - {s}^{ n- 1 }f $0$ - }$ $3$\displaystyle{\cdots - s{f}^{ $ n- 2 $ } $0$ - {f}^{ $ n- 1 $ } $0$ }$
	$3$\displaystyle{- tf $t$ }$	$3$\displaystyle{\frac{\hspace{2} dF $s$ \hspace{2}}{\hspace{2} ds \hspace{2}}}$
	$3$\displaystyle{{ $ - t $ }^{n}f $t$ }$	$3$\displaystyle{{F}^{ $n$ } $s$ }$
積分則	$3$\displaystyle{\int _{0}^{t} f $t$ dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}F $s$ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{f}^{ $ - 1 $ } $0$ }$
	$3$\displaystyle{ \int _{0}^{t} \int _{0}^{t} \cdots \int _{0}^{t} f $t$ { $ dt $ }^{n} }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{n} \hspace{2}}F $s$ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{n} \hspace{2}}{f}^{ $ - 1 $ } $0$ +}$ $3$\displaystyle{\cdots +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{f}^{ $ - n $ } $0$ }$
	$3$\displaystyle{\int _{ - \infty }^{t} f $t$ dt }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}} \[ \int _{ - \infty }^{t} f $t$ dt \] +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}F $s$ }$
	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ }$	$3$\displaystyle{\int _{s}^{\infty} F $s$ ds}$
	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {t}^{n} \hspace{2}}f $t$ }$	$3$\displaystyle{\int _{s}^{\infty} \int _{s}^{\infty} \cdots \int _{s}^{\infty} F $s$ { $ ds $ }^{n}}$
合成積則	$3$\displaystyle{\int _{0}^{t} \hspace{1}{f}_{1} $ t- {\tau} $ \hspace{1}{f}_{2} ${\tau}$ d{\tau}}$	$3$\displaystyle{\hspace{1}{F}_{1} $s$ \hspace{1}{F}_{2} $s$ }$
合成積則	$3$\displaystyle{\hspace{1}{f}_{1} $t$ \hspace{1}{f}_{2} $t$ }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi}j \hspace{2}}\hspace{1}{\int }_{ Br } \hspace{1}{F}_{1} $ s- {\sigma} $ \hspace{1}{F}_{2} ${\sigma}$ d{\sigma}}$
パラメータによる微分則	$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\alpha} \hspace{2}} \{ f $ t,{\alpha} $ \} }$	$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\alpha} \hspace{2}} \{ F $ s,{\alpha} $ \} }$
パラメータによる積分則	$3$\displaystyle{\int _{a}^{b} f $ t,{\alpha} $ d{\alpha}}$	$3$\displaystyle{\int _{a}^{b} F $ s,{\alpha} $ d{\alpha}}$
パラメータによる極限則	$3$\displaystyle{{ \lim }\limits_{ {\alpha}\rightarrow a }f $ t,{\alpha} $ }$	$3$\displaystyle{{ \lim }\limits_{ {\alpha}\rightarrow a }F $ s,{\alpha} $ }$
積分対応式	$3$\displaystyle{\int _{0}^{t} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}\int _{s}^{\infty} F $s$ ds }$
積分対応式	$3$\displaystyle{\int _{t}^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}\int _{0}^{s} F $s$ ds }$
積分等式	$3$\displaystyle{\int _{0}^{\infty} f $t$ dt}$	$3$\displaystyle{ \[ F $s$ \] _{\infty}^{0}}$
積分等式	$3$\displaystyle{\int _{0}^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f $t$ dt}$	$3$\displaystyle{\int _{0}^{\infty} F $s$ ds }$
初期値定理	$3$\displaystyle{{ \lim }\limits_{ t\rightarrow 0 }f $t$ }$	$3$\displaystyle{{ \lim }\limits_{ s\rightarrow \infty }sF $s$ }$
最終値定理	$3$\displaystyle{{ \lim }\limits_{ t\rightarrow \infty }f $t$ }$	$3$\displaystyle{ { \lim }\limits_{ s\rightarrow 0 }sF $s$ }$

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　最終更新日： 2023年6月1日

基本公式	$3$\displaystyle{f \(t\) }$ (実数 $3$\displaystyle{t}$ の関数)	$3$\displaystyle{F \(s\) }$ ( $3$\displaystyle{s}$ の関数)
線形性	$3$\displaystyle{ {\sum }\limits_{n} \hspace{1}{a}_{n} f \(t\) }$	$3$\displaystyle{ {\sum }\limits_{n} \hspace{1}{a}_{n} \hspace{1}{F}_{n} \(s\) }$
相似定理	$3$\displaystyle{f \( at \) }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}F \( \frac{\hspace{2}s\hspace{2}}{\hspace{2}a\hspace{2}} \) }$
相似定理	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}f \( \frac{\hspace{2}t\hspace{2}}{\hspace{2}a\hspace{2}} \) }$	$3$\displaystyle{F \( as \) }$
推移則	$3$\displaystyle{f \( t- a \) u \( t- a \) }$	$3$\displaystyle{{e}^{ - as }F \(s\) }$
推移則	$3$\displaystyle{{e}^{ - at }f \(t\) }$	$3$\displaystyle{F \( s+a \) }$
微分則	$3$\displaystyle{ \frac{\hspace{2} df \(t\) \hspace{2}}{\hspace{2} dt \hspace{2}} }$	$3$\displaystyle{ sF \(s\) - f \(0\) }$
	$3$\displaystyle{ \frac{\hspace{2} {d}^{2}f \(t\) \hspace{2}}{\hspace{2} d{t}^{2} \hspace{2}} }$	$3$\displaystyle{ {s}^{2}F \(s\) - sf \(0\) - {f}^{\prime } \(0\) }$
	$3$\displaystyle{ \frac{\hspace{2} {d}^{n}f \(t\) \hspace{2}}{\hspace{2} d{t}^{n} \hspace{2}} }$	$3$\displaystyle{ {s}^{n}F \(s\) - {s}^{ n- 1 }f \(0\) - }$ $3$\displaystyle{\cdots - s{f}^{ \( n- 2 \) } \(0\) - {f}^{ \( n- 1 \) } \(0\) }$
	$3$\displaystyle{- tf \(t\) }$	$3$\displaystyle{\frac{\hspace{2} dF \(s\) \hspace{2}}{\hspace{2} ds \hspace{2}}}$
	$3$\displaystyle{{ \( - t \) }^{n}f \(t\) }$	$3$\displaystyle{{F}^{ \(n\) } \(s\) }$
積分則	$3$\displaystyle{\int _{0}^{t} f \(t\) dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}F \(s\) +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{f}^{ \( - 1 \) } \(0\) }$
	$3$\displaystyle{ \int _{0}^{t} \int _{0}^{t} \cdots \int _{0}^{t} f \(t\) { \( dt \) }^{n} }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{n} \hspace{2}}F \(s\) +\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{n} \hspace{2}}{f}^{ \( - 1 \) } \(0\) +}$ $3$\displaystyle{\cdots +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}{f}^{ \( - n \) } \(0\) }$
	$3$\displaystyle{\int _{ - \infty }^{t} f \(t\) dt }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}} \[ \int _{ - \infty }^{t} f \(t\) dt \] +\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}F \(s\) }$
	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f \(t\) }$	$3$\displaystyle{\int _{s}^{\infty} F \(s\) ds}$
	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {t}^{n} \hspace{2}}f \(t\) }$	$3$\displaystyle{\int _{s}^{\infty} \int _{s}^{\infty} \cdots \int _{s}^{\infty} F \(s\) { \( ds \) }^{n}}$
合成積則	$3$\displaystyle{\int _{0}^{t} \hspace{1}{f}_{1} \( t- {\tau} \) \hspace{1}{f}_{2} \({\tau}\) d{\tau}}$	$3$\displaystyle{\hspace{1}{F}_{1} \(s\) \hspace{1}{F}_{2} \(s\) }$
合成積則	$3$\displaystyle{\hspace{1}{f}_{1} \(t\) \hspace{1}{f}_{2} \(t\) }$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi}j \hspace{2}}\hspace{1}{\int }_{ Br } \hspace{1}{F}_{1} \( s- {\sigma} \) \hspace{1}{F}_{2} \({\sigma}\) d{\sigma}}$
パラメータによる微分則	$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\alpha} \hspace{2}} \{ f \( t,{\alpha} \) \} }$	$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\alpha} \hspace{2}} \{ F \( s,{\alpha} \) \} }$
パラメータによる積分則	$3$\displaystyle{\int _{a}^{b} f \( t,{\alpha} \) d{\alpha}}$	$3$\displaystyle{\int _{a}^{b} F \( s,{\alpha} \) d{\alpha}}$
パラメータによる極限則	$3$\displaystyle{{ \lim }\limits_{ {\alpha}\rightarrow a }f \( t,{\alpha} \) }$	$3$\displaystyle{{ \lim }\limits_{ {\alpha}\rightarrow a }F \( s,{\alpha} \) }$
積分対応式	$3$\displaystyle{\int _{0}^{t} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f \(t\) dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}\int _{s}^{\infty} F \(s\) ds }$
積分対応式	$3$\displaystyle{\int _{t}^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f \(t\) dt}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}}\int _{0}^{s} F \(s\) ds }$
積分等式	$3$\displaystyle{\int _{0}^{\infty} f \(t\) dt}$	$3$\displaystyle{ \[ F \(s\) \] _{\infty}^{0}}$
積分等式	$3$\displaystyle{\int _{0}^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}f \(t\) dt}$	$3$\displaystyle{\int _{0}^{\infty} F \(s\) ds }$
初期値定理	$3$\displaystyle{{ \lim }\limits_{ t\rightarrow 0 }f \(t\) }$	$3$\displaystyle{{ \lim }\limits_{ s\rightarrow \infty }sF \(s\) }$
最終値定理	$3$\displaystyle{{ \lim }\limits_{ t\rightarrow \infty }f \(t\) }$	$3$\displaystyle{ { \lim }\limits_{ s\rightarrow 0 }sF \(s\) }$