Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same conditions, in $n$-dimensions, and look to get the $n$-dimensional analogue of d'Alembert's formula.
I feel comfortable doing this, but I'm stuck on what looks like a routine computation, and so I seek help getting past this step.
What I have done so far is take the Fourier transform of the wave equation, to get: $$\hat{u}(t, \xi) = A \cos{c|\xi|t} + B\sin{c|\xi|t}$$
subject to $\hat{u}(0, \xi) = \hat{f}(\xi)$ and $\hat{u_{t}}(\xi) = \hat{g}(\xi)$. I determined that $A = \frac{1}{2} \left(\hat{f} + \frac{\hat{g}}{c|\xi|} \right)$ and $B = \frac{1}{2} \left(\hat{f} - \frac{\hat{g}}{c|\xi|} \right)$.
At this point, I'm just wondering if there's anything left to do. That is, I will essentially solve for $u$ if I apply the inverse Fourier transform. But is the result something that can easily be computed, or is it better left in integral form?