It is known that $\int_{x\in S}\exp(\kappa\mu^Tx)dx$ where S is the surface of the unit sphere is $\frac{(2\pi)^{p/2} I_{p/2-1}(\kappa) }{\kappa^{p/2-1}}$ where $p$ is the number of dimensions and $I$ is the modified bessel function of the first kind: http://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution.
However, what would happen if I restrict this sphere to the first quadrant. i.e. $x_i>0$?
$$\int_{x\in S, x_i>0}\exp(\kappa\mu^Tx)dx=?$$