Inverse Fourier transform of $ \frac{\sin(c|\mathbf{k}|)}{c|\mathbf{k}|} $

Question

What is the inverse Fourier transform of $$ \tilde f(k) = \frac{\sin(ck)}{ck} $$ where $k =|\mathbf{k}|$ and $\mathbf{k} \in \mathbb R^n$. The inverse transform should be $$ f(\mathbf x) = \mathcal{F}^{-1}[\tilde f](\mathbf x)=\frac{1}{(2\pi)^\frac{n}{2}}\int_{\mathbb R^n}\frac{\sin(ck)}{ck} e^{i \mathbf x \cdot \mathbf k} dk_1\dots dk_n. $$ My attempt was based on considering $$\mathbf x = |\mathbf x|(\cos \alpha_1,...,\cos \alpha_n)\quad \text{and} \quad \mathbf k = |\mathbf k|(\cos \beta_1,...,\cos \beta_n) $$ so I get $$ \mathbf x \cdot \mathbf k = xk (\cos \alpha_1 \cos \beta_1,...,\cos \alpha_n\cos \beta_n) = xk\ \phi(\alpha,\beta) $$ where $$ \sum_{j=1}^n \cos^2(\alpha_j) = \sum_{j=1}^n \cos^2(\beta_j) = 1 $$ so they are related, and somehow $dk_1\dots dk_n$ becomes proportional to $k^{n-1}dk$, that is $$ f(\mathbf x) = \mathcal{F}^{-1}[\tilde f](\mathbf x) = \frac{1}{(2\pi)^\frac{n}{2}}\int_{[0,2\pi]^{n-1}}\left( \int_{\mathbb R^+}\frac{\sin(ck)}{ck} e^{i xk\ \phi(\alpha,\beta)} k^{n-1}dk \right) \psi(\beta)d\beta_1\dots\beta_{n-1} $$ My problem is the integral wrt $k$, since for 3-dimensional case I can use the spherical coordinates, so $\psi(\beta)$ is known. But I don't know yet how to proceed, this integral doesn't seem to converge.