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	$\150dpi \large \frac{F_{1}(s)}{1-e^{-sT}}$