RC circuit not ideal sampling
As OP has mentioned,
\$s(t) = \exp(-t/RC)\$ for \$0 &lt; t &lt; T_c\$, elsewhere zero. i.e. it can be re written as
\$\color{red}{s(t) = \exp(-t/RC) \cdot (u(t) - u(t - T_c))}\$
\$s(t) = \exp(-t/RC) \cdot u(t) - \exp(-(t-T_c + T_c)/RC) \cdot u(t - T_c)\$
\$s(t)= \exp(-t/RC) \cdot u(t) - \exp(-T_c/RC) \exp(-(t-T_c)/RC) \cdot u(t - T_c) \$. taking Laplace transform (and using time shifting theorem),
\$S(w) = \frac{1}{w+1/RC} - \exp(-T_c/RC) \cdot \frac{\exp(-T_c \cdot w)}{w+1/RC}\$
\$S(w) = \frac{1}{w+1/RC} (1 - \exp(-T_c/RC) \cdot \exp(-T_c w))\$
The above equation is what seems to be present on top of page 22.
Note that I have used \$w\$ as the Laplace variable since \$s\$ is already being used for another purpose.