\begin{nicebox}
{\bf Time-Shifting (Translation) Property }
\begin{align*}
x(t+T) &\ltp e^{sT}\hat{X}(s) \\
\end{align*}
The $RoC$ excludes $\infty$ if the signal is not causal, or excludes $-\infty$ if the signal is not anti-causal. 
\end{nicebox}

\begin{nicebox}
{\bf Convolution in the Time Domain }
\begin{align*}
h(t) &= f(t) * g(t) \\
\hat{H}(s) &= \hat{F}(s)\hat{G}(s) \\
\end{align*}
The $RoC(h) \supseteq RoC(f) \cap RoC(g)$.
\end{nicebox}

\begin{nicebox}
{\bf Modulation with a Complex Exponential }
\begin{align*}
e^{s_0 t} x(t) &\ltp \hat{X}(s-s_0) \\
\end{align*}
$RoC = RoC(x) + \Real{s_0} $
\end{nicebox}

\begin{nicebox}
{\bf Integration in the Time Domain }
\begin{align*}
 \int_{-\infty}^t x(\tau) d\tau &\ltp \frac{\hat{X}(s)}{s}
\end{align*}
$RoC(y) \supseteq RoC(x) \cap \{ s \st \Real{s} \gt 0 \} $
\end{nicebox}


\begin{nicebox}
{\bf Differentiation in the Time Domain }
\begin{align*}
\frac{d}{dt} x(t) \ltp s\hat{X}(s) \\
\end{align*}
$RoC \supseteq RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Generalized Differentiation in the Time Domain }
\begin{align*}
\frac{d^k}{dt^k} x(t) \ltp s^k \hat{X}(s) \quad k \in \N \\
\end{align*}
$RoC \supseteq RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Frequency Differentiation }
\begin{align*}
-tx(t) \ltp \frac{d}{ds}\hat{X}(s) \\
\end{align*}
$RoC = RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Conjugation Property }
\begin{align*}
x^*(t) &\ltp \hat{X}^*(s^*)
\end{align*}
$RoC = RoC(x)$
\end{nicebox}

Summary of Laplace Transforms