e의 At승의 라플라스 변환

$$e^{At}=I+tA+\frac{t^2}{2!}{A}^2+\frac{t^3}{3!}{A}^3+...$$

이때,

$$L\left[\frac{t^k}{k!}\right]={\int }_0^{\mathrm{\infty }}\frac{t^k}{k!}{e}^{-st}dt$$

$$=0+{\int }_0^{\mathrm{\infty }}\frac{1}{s^k}{e}^{-st}dt(\text{Applied partial integration})$$

$$=s^{-(k+1)}$$

이므로

$$L\left[e^{At}\right]=L[I]+L[tA]+L\left[\frac{t^2}{2!}{A}^2\right]+...$$

$$=s^{-1}I+s^{-2}A+s^{-3}{A}^2+...$$

$$=s^{-1}(I+s^{-1}{A}^1+s^{-2}{A}^2+...)$$

$$=s^{-1}\frac{1}{I-s^{-1}A},(s>A)$$

$$=s^{-1}(I-s^{-1}A{)}^{-1}$$

$$=(sI-A{)}^{-1}$$