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 linearity 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 Scaling Property
If $,x (t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$
Then time scaling 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
$int x (t) dt stackrel{mathrm{L.T}}{longleftrightarrow} {1 over s} X(s)$
$iiint ,…, int x (t) dt stackrel{mathrm{L.T}}{longleftrightarrow} {1 over s^n} X(s)$
Multiplication and Convolution Properties
If $,x(t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s)$
and $ y(t) stackrel{mathrm{L.T}}{longleftrightarrow} Y(s)$
Then multiplication 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
$x(t) * y(t) stackrel{mathrm{L.T}}{longleftrightarrow} X(s).Y(s)$
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