# How to integrate a function over an n-dimensional sphere with a given radius?

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!

• The first step is to rotate the coordinates so that $x = (0, 0, \dots, 1)$. May 8, 2023 at 16:12
• Sorry, what do you mean by rotating the coordinates? May 8, 2023 at 16:23