Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively.


I need to solve
$$\int\_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right) 
\, dx$$
where $(\cdot )^\ast$ is conjugate transpose of a vector and $\mathcal{R}\{ \cdot\}$ takes the real part.


Is this doable. Seems related to the Itzykson-zuber integral, but I cannot find anything in the literature.



integrate exponential function of quadratic form over unit norm vectors
I doubt that there is a closed-form expression for any $M$, but for large $M$ you can approximate the integral by replacing the constraint $\|x\|=1$ by a Gaussian measure  

uncorrelated $x\_n$'s with mean and variance $E(x\_n)=0$, $E(|x\_n|^2)=1/M$  

I assume that $A$ is positive definite, with eigenvalues $a\_n>0$, $n=1,2,\ldots M$. Note that the integral depends only on the eigenvalues of $A$, not on the eigenvectors, so we may as well take $A$ diagonal. The vector $b=(b\_1,b\_2,\ldots b\_M)$ is then taken in that basis. It is convenient to normalize the integral by the area $S\_M$ of the $M$-dimensional unit hypersphere.  

I then arrive at
$$\frac{1}{S\_M}\int\_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right)dx$$
$$=\frac{e^{b^\ast A^{-1}b}}{S\_M}\int\_{\|x\|^2=1}\exp[-(x-A^{-1}b)^\ast A(x-A^{-1}b)]\,dx\rightarrow$$
$$\rightarrow\prod\_{n=1}^M\left(e^{|b\_n|^2/a\_n}\int\_{-\infty}^\infty\exp[-(x-b\_n/a\_n)^\ast a\_n(x-b\_n/a\_n)]e^{-|x|^2/M}\,\frac{dx}{\pi M}\right)$$
$$=\prod\_{n=1}^M\frac{\exp\left(\frac{M|b\_n|^2}{1+Ma\_n}\right)}{1+Ma\_n}$$


