Let $ \vec{x}=\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} \in \mathbb{R}^3, |\vec{x}|=x $ and $ \vec{Y}=\begin{bmatrix} Y_1\\ Y_2\\ Y_3 \end{bmatrix} $
Consider an expression $ f(\vec{x})=e^{i \vec{x}\cdot\vec{Y}}=e^{i(x_1 Y_1+x_2 Y_2+x_3 Y_3)} $
Is there any general result concerning the form of the integral over the unit sphere: $$ \int_{S^2}d\vec{x} e^{i \vec{x}\cdot\vec{Y}} $$
It seems natural to express the problem in a spherical system $ \vec{x}=\begin{bmatrix} \sin{\theta}\sin{\phi}\\ \sin{\theta}\cos{\phi}\\ \cos{\theta} \end{bmatrix} $
Our integral in the spherical system is understood as follows $ \int_{S^2}d\vec{x} f(\vec{x}) = \int_{0}^{2\pi}d\phi \int_{0}^{\pi}d\theta \sin{\theta} f(\vec{x}(\theta,\phi)) $
$$ \int_{0}^{2\pi}d\phi \int_{0}^{\pi}d\theta \sin{\theta} e^{i(Y_1 \sin{\theta}\sin{\phi} +Y_2 \sin{\theta}\cos{\phi} +Y_3 \cos{\theta} )} $$
With respect to $\phi $ I manage to integrate and Jacobi functions come out. However, I can not cope with integration with respect to $\theta$.
Does any of the esteemed colleagues know the solution to this integral?
(I do not insist on spherical coordinates.)