## Properties of The Laplace Transform

Here are some important properties of the Laplace Transform F(s) being the Laplace transform of f(t).

 $\150dpi \large \frac{d}{dt}f(t)$ $\150dpi \large sF(S)-f(0)$ $\150dpi \large \frac{{d}^{2}}{dt^{2}}f(t)$ $\150dpi \large s^{2}F(S)-sf(0)-f'(0)$ $\150dpi \large \frac{{d}^{n}}{dt^{n}}f(t)$ $\150dpi \large s^{n}F(S)-\sum_{i=1}^{n}s^{(n-i)}f^{i-1}(0)$ $\150dpi \large \int_{0}^{t}f(\lambda )d\lambda$ $\150dpi \large \frac{F(s)}{s}$ $\150dpi \large tf(t)$ $\150dpi \large -\frac{dF(s)}{ds}$ $\150dpi \large \frac{f(t)}{t}$ $\150dpi \large \int_{s}^{\infty }F(s)ds$ $\150dpi \large {f(t-a)}{u(t-a)}$ $\150dpi \large F(s)e^{-as}$ $\150dpi \large e^{-at}{f(t)}$ $\150dpi \large F(s+a)$ $\150dpi \large {f(\frac{t}{a})}$ $\150dpi \large aF(as)$ Initial Value Theorem $\150dpi \large \lim_{t\to 0}f(t)=\lim_{s\to \infty }sF(s)$ Final Value Theorem $\150dpi \large \lim_{t\to \infty }f(t)=\lim_{s\to 0 }sF(s)$ $\120dpi f(t)$ periodic with a period T $\150dpi \large \frac{F_{1}(s)}{1-e^{-sT}}$

Above, F1(s) is the Laplace transform of f(t) for the first cycle.