In this topic, you study the Laplace Transform Properties as Linearity, Time Scaling, Time Shifting, Frequency Shifting, Time differentiation, Time integration, Time Reversal, Convolution in time and Multiplication in time.
Linearity
if
\[a{f_1}(t) \leftrightarrow a{F_1}(s)\]
\[b{f_2}(t) \leftrightarrow b{F_2}(s)\]
Then
\[a{f_1}(t){\text{ }}+{\text{ }}b{f_2}(t) \leftrightarrow a{F_1}(s){\text{ }} + {\text{ }}b{F_2}(s)\]
Time Scaling
if
\[f(t) \leftrightarrow F(s)\]
Then
\[f(at) \leftrightarrow \frac{1}{{\left| a \right|}}F\left( {\frac{s}{a}} \right)\]
Time Shifting
if
\[f(t) \leftrightarrow F(s)\]
Then
\[f(t – {t_0}) \leftrightarrow F({s}){e^{ – s {t_0}}}\]
Frequency Shifting
if
\[f(t) \leftrightarrow F(s )\]
Then
\[f(t){e^{{a}t}} \leftrightarrow F(s{} – {a})\]
Time differentiation
if
\[f(t) \leftrightarrow F(s)\]
Then
\[\frac{d}{{dt}}\left[ {f(t)} \right] \leftrightarrow s F(s)\]
Time integration
if
\[f(t) \leftrightarrow F(s)\]
Then
\[\int\limits_{ – \infty }^t {f(\tau )d\tau } \leftrightarrow \frac{1}{{s}}F(s)\]
Time Reversal
if
\[f(t) \leftrightarrow F(s)\]
Then
\[f( – t) \leftrightarrow F( – s )\]
Convolution in time
if
\[{f_1}(t) \leftrightarrow {F_1}(s)\]
\[{f_2}(t) \leftrightarrow {F_2}(s)\]
Then
\[{f_1}(t)*{f_2}(t) \leftrightarrow {F_1}(s){F_2}(s)\]
Multiplication in time
if
\[{f_1}(t) \leftrightarrow {F_1}(\omega )\]
\[{f_2}(t) \leftrightarrow {F_2}(\omega )\]
Then
\[{f_1}(t){f_2}(t) \leftrightarrow \frac{1}{{2\pi j}}\left[ {{F_1}(s)*{F_2}(s )} \right]\]
