Laplace変換・逆変換　Laplace変換の例6

Laplace変換・逆変換 電験 理論

Laplace変換の微分

\displaystyle{F(s)=\int_0^\infty f(t) \exp(-st)dt}

の両辺をsで微分すると

\displaystyle{ \frac{d}{ds} F(s) = \frac{d}{ds} \int_0^\infty f(t) \exp(-st) dt = - \int_0^\infty t f(t) \exp(-st) dt }

故に

\displaystyle{\mathcal{L}[t f(t)] (s)= - \frac{d F(s)}{ds}}

 

\displaystyle{ \mathcal{L}[t](s)= -\frac{d}{ds} \frac{1}{s} = \frac{1}{s^2} }

\displaystyle{\mathcal{L}[t^2](s)=-\frac{d}{ds}\frac{1}{s^2} = \frac{2}{s^3}}

\displaystyle{ \mathcal{L}[t^n](s)=\frac{(n-1)!}{s^n}}