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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 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
$x (at) \stackrel{\mathrm{L.T}}{\longleftrightarrow} {1\over |a|} X({s\over a})$
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
$x(t) * y(t) \stackrel{\mathrm{L.T}}{\longleftrightarrow} X(s).Y(s)$