Laplace Transform

Topics

• Laplace Transform Properties
• Laplace Transform Catalog
• Partial Fraction Decomposition

$\mathcal{L}\left[f(t)\right]=F(s)={\int }_0^{\infty }{e}^{-st}f(t)\mathrm{d}t$

• Unilateral Laplace transform
• $s=\sigma +j\omega$
• Bilateral version includes a $-\infty$ lower bound

$f(t)=\frac{1}{2\pi j}{\int }_{c-j\omega }^{c+j\omega }F(s)e^{st}\mathrm{d}s$

• Inverse bilateral Laplace transform
• Assumes $c>{\sigma }_0$

$F(j\omega )=F(s){|}_{s=j\omega }$

• The Fourier Transform is a special case of the Laplace Transform