反復積分に関するコーシーの公式

\begin{eqnarray} \mathcal{L}_t\Big[\int_0^tf(\sigma)\mathrm{d}\sigma\Big](s) &=&\mathcal{L}_{t}\Big[u(t)*f(t)\Big](s) \\ &=&\frac{F(s)}{s}　\tag{*} \end{eqnarray}

$$ \mathcal{L}_t\Big[\int_{a}^{t}\int_{a}^{\sigma_1}\cdots\int_{a}^{\sigma_{n-1}}f(\sigma_{n})\mathrm{d}\sigma_n\cdots\mathrm{d}\sigma_2\mathrm{d}\sigma_1\Big](s) $$

\begin{eqnarray} &=&\mathcal{L}_{t}\Big[\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\Big](s) \quad\big(\sigma_i-a\mapsto\sigma_i \quad(i=1,...,n)\big) \\ &=&e^{-sa}\mathcal{L}_{t}\Big[\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\Big](s) \quad(\text{Laplace変換の遷移則}) \\ &=&\frac{e^{-sa}}{s^n}\mathcal{L}_{\sigma_n}\big[f(\sigma_{n}+a)\big](s) \quad(\because\text{*をn回適用}) \end{eqnarray}
次のアスタリスクは合成積です。
\begin{eqnarray} \mathcal{L}_{s}^{-1}\Big[\frac{e^{-sa}}{s^n}\mathcal{L}_{t}\big[f(t+a)\big](s)\Big](t) &=&\frac{1}{\Gamma(n)}\mathcal{L}_{s}^{-1}\Big[\frac{\Gamma(n)}{s^n}\Big](t)*\mathcal{L}_{s}^{-1}\Big[e^{-sa}\mathcal{L}_{t}\big[f(t+a)\big](s)\Big](t) \\ &=&\frac{1}{(n-1)!}t^{n-1}*\big(f(t)u(t-a)\big) \\ &=&\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{eqnarray}