Find Spherical Integral: $\int_{\| {\bf x}\|=1} x_1 e^{ {\bf x}^T{\bf y}} {\rm d} {\bf x}$

I am interested in the following integral: given a vector ${\bf y} \in \mathbb{R}^n$ find \begin{align} \int_{\| {\bf x}\|=1} x_1 e^{ {\bf x}^T{\bf y}} {\rm d} {\bf x} \end{align} where $x_1$ denotes the first coordinate of ${\bf x}$. In other words, we are integrating over a sphere of unit sphere.

After searching around I found that the following integral: \begin{align} \int_{\| {\bf x}\|=1} e^{ {\bf x}^T{\bf y}} {\rm d} {\bf x}= \left( \frac{\|{\bf y}\|}{2} \right)^{1-\frac{n}{2}} S_{n-1} \Gamma(n/2) I_{n/2-1}(\| {\bf y} \|) \end{align} where $S_{n-1}$ is the volume of $n-1$-sphere and $I_v$ is modified Bessel function of the first kind.

I think this result can be of use for us. In particular, I was thinking to integrate for $n-2$ sphere fist and then over $x_1$ coordinate

Solution 1:

There is no more work to be done. The answer is the derivative of the expression you found w.r.t. $y_1$

$$\int_{|x|=1}x_1\exp(x\cdot y)dx = \frac{\partial}{\partial y_1}\int_{|x|=1}\exp(x\cdot y)dx = \frac{\partial}{\partial y_1}\left[\left( \frac{|y|}{2} \right)^{1-\frac{n}{2}} S_{n-1} \Gamma(n/2) I_{n/2-1}(|y|)\right]$$