= Solution
Integrating the complex exponential and taking its even and odd combinations gives the standard <Laplace transforms>
$$
\boxed{\mathcal L[a_1](p)=\frac p{p^2+\omega^2},\qquad
\mathcal L[a_2](p)=\frac\omega{p^2+\omega^2}.}
$$
For example, $\mathcal L[e^{i\omega t}]=(p-i\omega)^{-1}$ for $\operatorname{Re}p>0$, and the real sine and cosine identities follow by combining the two signs of $i\omega$.
Apply part (a) once to the cosine transform and twice to the sine transform:
$$
\begin{aligned}
\mathcal L[a_3](p)&=-\frac d{dp}\frac p{p^2+\omega^2}
=\boxed{\frac{p^2-\omega^2}{(p^2+\omega^2)^2}},\\
\mathcal L[a_4](p)&=\frac{d^2}{dp^2}\frac\omega{p^2+\omega^2}
=\boxed{\frac{2\omega(3p^2-\omega^2)}{(p^2+\omega^2)^3}}.
\end{aligned}
$$
All four transforms are valid on the stated right half-plane.
Back to article page
