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ラプラス変換表

よく用いられる基本関数のラプラス変換についてまとめています．

詳細は各公式のページを参照してください．

$3$\displaystyle{f \(t\) }$ (実数 $3$\displaystyle{t}$ の関数)	$3$\displaystyle{F \(s\) }$ ( $3$\displaystyle{s}$ の関数)
$3$\displaystyle{K}$	$3$\displaystyle{\frac{\hspace{2}K\hspace{2}}{\hspace{2}s\hspace{2}}}$
$3$\displaystyle{Kt}$	$3$\displaystyle{\frac{\hspace{2}K\hspace{2}}{\hspace{2} {s}^{2} \hspace{2}}}$
$3$\displaystyle{{\delta} \(t\) }$	$3$\displaystyle{1}$
$3$\displaystyle{ u \(t\) }$	$3$\displaystyle{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}s\hspace{2}} }$
$3$\displaystyle{t}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{2} \hspace{2}}}$
$3$\displaystyle{{t}^{n}}$	$3$\displaystyle{\frac{\hspace{2} n! \hspace{2}}{\hspace{2} {s}^{ n+1 } \hspace{2}}}$
$3$\displaystyle{{t}^{a}}$	$3$\displaystyle{\frac{\hspace{2} {\Gamma} \( {\alpha}+1 \) \hspace{2}}{\hspace{2} {s}^{ {\alpha}+1 } \hspace{2}}}$
$3$\displaystyle{{e}^{ - at }}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} s+a \hspace{2}}}$
$3$\displaystyle{t{e}^{ - at }}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \( s+a \) }^{2} \hspace{2}}}$
$3$\displaystyle{{t}^{n}{e}^{ - at }}$	$3$\displaystyle{ \frac{\hspace{2} n! \hspace{2}}{\hspace{2} { \left({ s+a }\right) }^{ n+1 } \hspace{2}} }$
$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ {\pi}t } \hspace{2}}}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{s} \hspace{2}}}$
$3$\displaystyle{\sqrt{ \frac{\hspace{2} 4t \hspace{2}}{\hspace{2}{\pi}\hspace{2}} }}$	$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {s}^{ \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } \hspace{2}}}$
$3$\displaystyle{\sin {\omega}t}$	$3$\displaystyle{\frac{\hspace{2}{\omega}\hspace{2}}{\hspace{2} {s}^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{\cos {\omega}t}$	$3$\displaystyle{\frac{\hspace{2}s\hspace{2}}{\hspace{2} {s}^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{{e}^{ - at }\sin {\omega}t}$	$3$\displaystyle{\frac{\hspace{2}{\omega}\hspace{2}}{\hspace{2} { \( s+a \) }^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{{e}^{ - at }\cos {\omega}t}$	$3$\displaystyle{\frac{\hspace{2} s+a \hspace{2}}{\hspace{2} { \( s+a \) }^{2}+{{\omega}}^{2} \hspace{2}}}$

　

$3$\displaystyle{f \(t\) }$ (実数 $3$\displaystyle{t}$ の関数)	$3$\displaystyle{F \(s\) }$ ( $3$\displaystyle{s}$ の関数)
$3$\displaystyle{t{e}^{ - at }\sin {\omega}t}$	$3$\displaystyle{\frac{\hspace{2} 2{\omega} \( s+a \) \hspace{2}}{\hspace{2} { \[ { \( s+a \) }^{2}+{{\omega}}^{2} \] }^{2} \hspace{2}}}$
$3$\displaystyle{t{e}^{ - at }\cos {\omega}t}$	$3$\displaystyle{\frac{\hspace{2} { \( s+a \) }^{2}- {{\omega}}^{2} \hspace{2}}{\hspace{2} { \[ { \( s+a \) }^{2}+{{\omega}}^{2} \] }^{2} \hspace{2}}}$
$3$\displaystyle{t\sin {\omega}t}$	$3$\displaystyle{\frac{\hspace{2} 2{\omega}s \hspace{2}}{\hspace{2} { \( {s}^{2}+{{\omega}}^{2} \) }^{2} \hspace{2}}}$
$3$\displaystyle{t\cos {\omega}t}$	$3$\displaystyle{\frac{\hspace{2} {s}^{2}- {{\omega}}^{2} \hspace{2}}{\hspace{2} { \( {s}^{2}+{{\omega}}^{2} \) }^{2} \hspace{2}}}$
$3$\displaystyle{{\sin }^{2}{\omega}t}$	$3$\displaystyle{\frac{\hspace{2} 2{{\omega}}^{2} \hspace{2}}{\hspace{2} s \( {s}^{2}+4{{\omega}}^{2} \) \hspace{2}}}$
$3$\displaystyle{{\cos }^{2}{\omega}t}$	$3$\displaystyle{\frac{\hspace{2} {s}^{2}+2{{\omega}}^{2} \hspace{2}}{\hspace{2} s \( {s}^{2}+4{{\omega}}^{2} \) \hspace{2}}}$
$3$\displaystyle{\sin \( {\omega}t+{\theta} \) }$	$3$\displaystyle{\frac{\hspace{2} {\omega}\cos {\theta}+s\sin {\theta} \hspace{2}}{\hspace{2} {s}^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{\cos \( {\omega}t+{\theta} \) }$	$3$\displaystyle{\frac{\hspace{2} s\cos {\theta}- {\omega}\sin {\theta} \hspace{2}}{\hspace{2} {s}^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{{e}^{ - at }\sin \( {\omega}t+{\theta} \) }$	$3$\displaystyle{\frac{\hspace{2} \( s+a \) \sin {\theta}+{\omega}\cos {\theta} \hspace{2}}{\hspace{2} { \( s+a \) }^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{{e}^{ - at }\cos \( {\omega}t+{\theta} \) }$	$3$\displaystyle{\frac{\hspace{2} \( s+a \) \cos {\theta}- {\omega}\sin {\theta} \hspace{2}}{\hspace{2} { \( s+a \) }^{2}+{{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{\sin h{\omega}t}$	$3$\displaystyle{\frac{\hspace{2}{\omega}\hspace{2}}{\hspace{2} {s}^{2}- {{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{\cos h{\omega}t}$	$3$\displaystyle{\frac{\hspace{2}s\hspace{2}}{\hspace{2} {s}^{2}- {{\omega}}^{2} \hspace{2}}}$
$3$\displaystyle{{\sin h}^{2}{\omega}t}$	$3$\displaystyle{\frac{\hspace{2} 2{{\omega}}^{2} \hspace{2}}{\hspace{2} s \( {s}^{2}- 4{{\omega}}^{2} \) \hspace{2}}}$
$3$\displaystyle{{\cos h}^{2}{\omega}t}$	$3$\displaystyle{\frac{\hspace{2} {s}^{2}- 2{{\omega}}^{2} \hspace{2}}{\hspace{2} s \( {s}^{2}- 4{{\omega}}^{2} \) \hspace{2}}}$

　

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最終更新日： 2024年8月24日