I have been running into the following integral again and again:
Let $S^{n-1}= \{x \in \mathbb{R}^{n} \: | \: ||x||=1 \}$ and let $\lambda_{S^{n-1}}$ denote the surface measure over $S^{n-1}$ as defined in Stroock (2000) page 86.
Consider a fixed symmetric, positive definite matrix $Q$ of dimension $n \times n$, and a fixed scalar $a\in \mathbb{R}_{+}$
Question 1) Do you know if there is a closed form solution for the integral:
$$\int_{S^{n-1}} \exp\Big(a \omega'Q\omega \Big) \lambda_{S^{n-1}} (d \omega) $$
When $n=2$, I can express this integral as a modified Bessel function of the first kind $I_{v}(x)$ with $v=0$ evaluated at the eigenvalues of $Q$.
Question 2) Any suggestion about good numerical method for solving this integral?
Thanks!
*Stroock (2000) "A concise introduction to the theory of integration"