I am trying to evaluate the following integral:
$$\int_{S^n(\sigma)} e^{kx^Ty}dy$$
where $S^n(\sigma)$ is the sphere of radius $\sigma$ in $\mathbb{R}^n$, $k$ is a scalar, $x$ is a fixed vector in $\mathbb{R}^n$, and $y$ is the variable vector in $\mathbb{R}^n$.
I have not much experience in integrating over surfaces. I found a result that suggests that if k > 0 ,||x|| = 1 and we are integrating over an n-sphere of radius 1,the integral simplifies to $$\Gamma\left(\frac{n}{2}\right)\left(\frac{k}{2}\right)^{1-\frac{n}{2}}I_{\frac{n}{2}-1}(k)$$, where I is the modified Bessel function of first kind. This is in the book Directional Statistics by Mardia and Jupp at page 168. Is there a more general form or result for this kind of integral? Any hints or references would be appreciated. Thanks a lot!