I have two wave modes in moving frames
$a(s,z) = A(s + z/c,z)$ and $b(s,z) = B(s - z/c,z).$
I define
$\hat a(\omega,z) = \int e^{i\omega s}a(s,z)ds$ and $\hat b(\omega,z) = \int e^{i\omega s}b(s,z)ds$ where $t = s + z/c.$
I can show that
$$\begin{align}\int e^{i\omega s}\frac{\partial A}{\partial s}(s + z/c,z)\,ds = -i\omega\int a(s,z)ds \end{align}$$
since
$$\begin{align}\frac{\partial A}{\partial s} = \frac{\partial a}{\partial s} = -i\omega \hat a(\omega,s)\end{align}.$$
But I am having trouble now computing
$$\begin{align} \int e^{i\omega s} \frac{\partial A}{\partial t}(s + z/c,z)\,ds \end{align}$$
Could someone offer a hint please?