| Linearity |
$af(t)~+$ $bg(t)$ |
$aF(s)~+$ $bG(s)$ |
$a,b$ are constant |
| Scale change |
$f(at)$ |
$\dfrac{1}{a}F\left(\dfrac{s}{a}\right)$ |
$a$ > 0 |
| Shift |
$e^{-at}f(t)$ |
$F(s + a)$ |
|
| Delay |
$f(t-a)$ |
$e^{-as}F(s)$ |
|
| Derivation |
$\dfrac{df(t)}{dt}$ |
$sF(s) - f(0)$ |
|
| N-th derivation |
$\dfrac{d^nf(t)}{dt^n}$ |
$s^n~f(s)~-$ $s^{n-1}f(0)$ $~-s^{n-2}f^1(0)~-$ $\cdot\cdot\cdot$ $~-f^{n-1}(0)$ |
|
| Power |
$t^{n}f(t)$ |
$(-1)^n\dfrac{d^nF(s)}{ds^n}$ |
|
| Integration |
$\displaystyle \int_{0}^{t}f(x)dx$ |
$\dfrac{1}{s}F(s)$ |
|
| Reciprocal |
$\dfrac{1}{t}f(t)$ |
$\displaystyle \int_{s}^{\infty }F(x)dx$ |
|
| Convolution |
$f(t) * g(t)$ |
$F(s)\cdot G(s)$ |
* is the convolution operator |
| Periodic function |
$f(t)= f(t+T)$ |
$\dfrac{1}{1-e^{-sT}} \displaystyle \int_{0}^{T}e^{-sx}f(x)dx$ |
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