冪凾数のラプラス変換

ラプラス変換

$$\begin{array}{rcl}\mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}f\left(t\right)e^{–st}\mathrm{d}t\end{array}$$

$t^n$のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& t^n\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}{t}^n{e}^{–st}\mathrm{d}t\\ & =& {[t^n\cdot \frac{–1}{s}{e}^{-st}]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}n{t}^{n-1}\cdot \frac{–1}{s}{e}^{–st}\mathrm{d}t\\ & & \cdots {\int }_a^b{f}^{\prime }\left(t\right)g\left(t\right)\mathrm{d}t={\left[f\left(t\right)g\left(t\right)\right]}_a^b-{\int }_a^{b}f\left(t\right)g^{\prime }\left(t\right)\mathrm{d}t(f^{\prime }:f\mathrm{の}\mathrm{微}\mathrm{分},g^{\prime }:g\mathrm{の}\mathrm{部}\mathrm{分})\\ & & \cdots \int e^{at}\mathrm{d}t=\frac{1}{a}{e}^{at}+C\\ & & \cdots \frac{\mathrm{d}}{\mathrm{d}t}{t}^n=n{t}^{n-1}\\ & =& [{\mathrm{\infty }}^n\cdot \frac{–1}{s}{e}^{-s\mathrm{\infty }}-0^n\cdot \frac{–1}{s}{e}^{-s0}]-\left(\frac{–n}{s}\right){\int }_0^{\mathrm{\infty }}{t}^{n-1}{e}^{–st}\mathrm{d}t\\ & =& [0-0]+\frac{n}{s}{\int }_0^{\mathrm{\infty }}{t}^{n-1}{e}^{–st}\mathrm{d}t\\ & & \cdots e^{-\mathrm{\infty }}=0,\sin \left(0\right)=0\\ & =& \frac{n}{s}{\int }_0^{\mathrm{\infty }}{t}^{n-1}{e}^{–st}\mathrm{d}t\\ & =& \frac{n}{s}\frac{n-1}{s}{\int }_0^{\mathrm{\infty }}{t}^{n-2}{e}^{–st}\mathrm{d}t\\ & & ⋮\\ & =& \frac{n}{s}\frac{n-1}{s}\cdots \frac{2}{s}{\int }_0^{\mathrm{\infty }}t{e}^{–st}\mathrm{d}t\\ & =& \frac{n}{s}\frac{n-1}{s}\cdots \frac{2}{s}\frac{1}{s}{\int }_0^{\mathrm{\infty }}{e}^{–st}\mathrm{d}t\\ & =& \frac{n}{s}\frac{n-1}{s}\cdots \frac{2}{s}\frac{1}{s}{\left[\frac{–1}{s}{e}^{-st}\right]}_0^{\mathrm{\infty }}\\ & =& \frac{n!}{s^n}\frac{–1}{s}[e^{-s\mathrm{\infty }}-e^{-s0}]\\ & & \cdots n(n-1)\cdots 21=n!\\ & =& \frac{n!}{s^n}\frac{–1}{s}[0-1]\\ & & \cdots e^{-\mathrm{\infty }}=0,e^0=1\\ & =& \frac{n!}{s^n}\frac{–1}{s}(-1)\\ & =& \frac{n!}{s^{n+1}}\end{array}$$