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ラプラス変換基本公式表

 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 \( \frac{\hspace{2}s\hspace{2}}{\hspace{2}a\hspace{2}} \) }$ $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 \(s\) }$ $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 \(0\) - \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}^{ \( - 1 \) } \(0\) +\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} \(s\) }$ $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 }$ $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}}$ $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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初版：2009年3月4日，最終更新日： 2010年2月25日