e의 At승의 라플라스 변환

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

이때,

$$L\left[\frac{t^k}{k!}\right]=\int_0^\infty \frac{t^k}{k!}e^{-st}dt$$

$$=0+\int_0^\infty \frac{1}{s^k}e^{-st}dt \quad (\text{Applied partial integration})$$

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

이므로

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

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

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

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

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

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