I wish to verify that
$$\phi(r,t) = \int_r^\infty \frac{f_1(ct-R)}{\sqrt(R^2-r^2)}dR + \int_r^\infty \frac{f_2(ct+R)}{\sqrt(R^2-r^2)}dR,$$
where $f_1$ and $f_2$ are arbitrary functions, satisfies the axially symmetric wave equation,
$$\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial \phi}{\partial r}) - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = 0.$$
This solution is given in Section 71 of Fluid Mechanics by Landau and Lifschitz (Vol 6).
I am unsure how to take the derivatives with respect to $r$ and $t$ of the given solution $\phi$, ie my question is, "What are $\frac{\partial \phi}{\partial r}$ and $\frac{\partial \phi}{\partial t}$?"
I am aware of the Leibniz Integral Rule, however my issue here is that that the upper bound on both integrals is $\infty$, as well as the fact that evaluating the integrands at $r$ is undefined...
Any help is appreciated. Thank you!
EDIT: I made a mistake and the PDE I gave is not solved by the given function $\phi$. I misunderstood what the textbook was saying. Sorry!