Laplace Transforms Properties

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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

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