Laplace Transforms Properties

The properties of Laplace transform are:

Linearity Property

If $,x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

& $, y(t) stackrel{mathrm{L.T}}{longleftrightarrow} Y(s)$

Then pnearity property states that

$a x (t) + b y (t) stackrel{mathrm{L.T}}{longleftrightarrow} a X(s) + b Y(s)$

Time Shifting Property

If $,x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

Then time shifting property states that

$x (t-t_0) stackrel{mathrm{L.T}}{longleftrightarrow} e^{-st_0 } X(s)$

Frequency Shifting Property

If $, x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

Then frequency shifting property states that

$e^{s_0 t} . x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s-s_0)$

Time Reversal Property

If $,x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

Then time reversal property states that

$x (-t) stackrel{mathrm{L.T}}{longleftrightarrow} X(-s)$

Time Scapng Property

If $,x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

Then time scapng property states that

$x (at) stackrel{mathrm{L.T}}{longleftrightarrow} {1over |a|} X({sover a})$

Differentiation and Integration Properties

If $, x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

Then differentiation property states that

$ {dx (t) over dt} stackrel{mathrm{L.T}}{longleftrightarrow} s. X(s) - s. X(0) $

${d^n x (t) over dt^n} stackrel{mathrm{L.T}}{longleftrightarrow} (s)^n . X(s)$

The integration property states that

$int x (t) dt stackrel{mathrm{L.T}}{longleftrightarrow} {1 over s} X(s)$

$iiint ,..., int x (t) dt stackrel{mathrm{L.T}}{longleftrightarrow} {1 over s^n} X(s)$

Multippcation and Convolution Properties

If $,x(t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$

and $ y(t) stackrel{mathrm{L.T}}{longleftrightarrow} Y(s)$

Then multippcation property states that

$x(t). y(t) stackrel{mathrm{L.T}}{longleftrightarrow} {1 over 2 pi j} X(s)*Y(s)$

The convolution property states that

$x(t) * y(t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s).Y(s)$