反復積分に関するコーシーの公式

$$\begin{array}{rcl}{\mathcal{L}}_t[{\int }_0^{t}f(\sigma )\mathrm{d}\sigma ](s)& =& {\mathcal{L}}_t[u(t)\ast f(t)](s)\\ \text{(*)}& & =& \frac{F(s)}{s} \end{array}$$

$${\mathcal{L}}_t[{\int }_a^t{\int }_a^{{\sigma }_1}\cdots {\int }_a^{{\sigma }_{n-1}}f({\sigma }_n)\mathrm{d}{\sigma }_n\cdots \mathrm{d}{\sigma }_2\mathrm{d}{\sigma }_1](s)$$

$$\begin{array}{rcl}& =& {\mathcal{L}}_t[{\int }_0^{t-a}{\int }_0^{{\sigma }_1}\cdots {\int }_0^{{\sigma }_{n-1}}f({\sigma }_n+a)\mathrm{d}{\sigma }_n\cdots \mathrm{d}{\sigma }_2\mathrm{d}{\sigma }_1](s)({\sigma }_i-a\mapsto {\sigma }_i(i=1,...,n))\\ & =& e^{-sa}{\mathcal{L}}_t[{\int }_0^t{\int }_0^{{\sigma }_1}\cdots {\int }_0^{{\sigma }_{n-1}}f({\sigma }_n+a)\mathrm{d}{\sigma }_n\cdots \mathrm{d}{\sigma }_2\mathrm{d}{\sigma }_1](s)(\text{Laplace変換の遷移則})\\ & =& \frac{e^{-sa}}{s^n}{\mathcal{L}}_{{\sigma }_n}[f({\sigma }_n+a)](s)(\because \text{*をn回適用})\end{array}$$
次のアスタリスクは合成積です。
$$\begin{array}{rcl}{\mathcal{L}}_s^{-1}[\frac{e^{-sa}}{s^n}{\mathcal{L}}_t[f(t+a)](s)](t)& =& \frac{1}{\mathrm{\Gamma }(n)}{\mathcal{L}}_s^{-1}[\frac{\mathrm{\Gamma }(n)}{s^n}](t)\ast {\mathcal{L}}_s^{-1}[e^{-sa}{\mathcal{L}}_t[f(t+a)](s)](t)\\ & =& \frac{1}{(n-1)!}{t}^{n-1}\ast (f(t)u(t-a))\\ & =& \frac{1}{(n-1)!}{\int }_0^t(t-\tau {)}^{n-1}f(\tau )u(\tau -a)\mathrm{d}\tau \\ & =& \frac{1}{(n-1)!}{\int }_a^t(t-\tau {)}^{n-1}f(\tau )\mathrm{d}\tau \end{array}$$