An integral has been pushed me over the edge for several weeks. It reads as:
$$\displaystyle\int\_{\mathbb{R}\_y^3}\int\_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$


I tried to calculate the surface integral inside using spherical coordinates, but it seems that I couldn't do any further calculation since the integrand function is something like
$$e^{-\big(k\_1(\varphi)\sin^2\theta+k\_2(\varphi)\cos^2\theta+k\_3(\varphi)\sin\theta\cos\theta\big)}\sin\theta .$$
Then I tried to use variable substitution to compute, similarly, I didn't get anything useful. I was also trying to use Maple to compute, but it didn't work at all. My original intention is to prove that the formula
$$\displaystyle e^{-\frac{1}{2}|x|^2}\int\_{\mathbb{R}\_y^3}\int\_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$
is bounded.


I would be grateful if you could give me a definite result.



About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$
Without loss of generality you can orient the axes so that the vector $\mathbf{x}$ is along the $x\_3$ axis. Using also that $|\mathbf{\omega}|=1$, one has
$$|\mathbf{x}-[(\mathbf{x}-\mathbf{y})\cdot\mathbf{\omega}]\mathbf{\omega}|^2=(\mathbf{\omega}\cdot\mathbf{y})^2-x\_3^2(\omega\_3^2-1).$$
Subtitution into the integral
$$I=\displaystyle\int\_{\mathbb{R}\_y^3}\int\_{\mathbb{S}^2}e^{-\frac{1}{2}|\mathbf{x}-[(\mathbf{x}-\mathbf{y})\cdot\omega]\omega|^2}d\omega d\mathbf{y}$$
gives
$$I=\displaystyle\int\_{\mathbb{R}\_y^3}\int\_{\mathbb{S}^2}\exp\left[-\tfrac{1}{2}(\omega\cdot \mathbf{y})^2+\tfrac{1}{2}\left(\omega\_3^2-1\right) x\_3^2\right]\,d\omega d\mathbf{y}$$
$$=(2\pi)^{3/2}\int\_{-\infty}^\infty dy\_2\int\_{-\infty}^\infty dy\_3\int\_0^\pi d\theta\,\frac{\sin\theta}{|\cos\theta|}e^{-\tfrac{1}{2}x\_3^2\sin^2\theta}.$$
All three integrations are divergent.


