$\int_{S^{n-1}}\operatorname e^{ix\cdot \omega}\, \operatorname d\omega$

Question

Given $x \in \mathbb R^n$, there exists a simpler expresion for the integral? $$\int_{S^{n-1}}\operatorname e^{ix\cdot \omega}\, \operatorname d\omega$$ where $S^{n-1}$ is the sphere of $\mathbb R^n$.I know that only depends on |x| but nothing else.

Answer 1

Let $R\in SO(n)$ be a rotation such that $Rx=e^1=(|x|,0,\ldots,0)$, then the change of variables $y=R^{-1}\omega$ gives us $$\int_{S^{n-1}}e^{ix\cdot\omega}d\omega = \int_{S^{n-1}}e^{i|x|y_1}dy$$ This proves that the integral depends only from $|x|$. Now we can do another substitution by writing $$y = (\sin\phi,z\cos\phi)$$ where $\phi\in[-\pi/2,\pi/2]$ and $z\in S^{n-2}$ (where we assume $n\ge3$). This change of variables should give us (but I'd like a second opinion) $dy = \cos^{n-1}(\phi)d\phi dz$, and thus $$\int_{S^{n-1}}e^{i|x|y_1}dy=\int_{S^{n-2}}dz\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}e^{i|x|\phi}\cos^{n-1}(\phi)d\phi$$ It is well known that $$\int_{S^{n-2}}dz=\frac{2\pi^{(n-1)/2}}{\Gamma((n-1)/2)}$$ and the second integral should be solvable exactly (probably the best way is contour integration).