Laplace変換・逆変換　Laplace変換の例3

Laplace変換・逆変換 電験 理論

\delta関数と階段関数uのLaplace変換

\displaystyle{\mathcal{L}[\delta(t-a)](s)=\int_0^\infty \delta(t-a)\exp(-st)dt =\exp(-as)}

\displaystyle{\mathcal{L}[u(t-a)](s)=\int_0^\infty u(t-a)\exp(-st)dt }

\displaystyle{=\int_0^a 0 dt + \int_a^\infty \exp(-st)dt =\left[ - \frac{1}{s} \exp(-st) \right]_a^\infty = \frac{\exp(-as)}{s}}

\displaystyle{\mathcal{L}[u(t-a)f(t-a)](s)=\int_a^\infty f(t-a)\exp(-st)dt }

\displaystyle{=\int_0^\infty f(x) \exp(-s(a+x))dx = \exp(-sa) F(s)}