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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．

 $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}1\hspace{2}}{\hspace{2} { \( s+a \) }^{ 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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初版：2009年3月4日，最終更新日： 2009年6月23日