Selberg integral

Remarks

For $n = 1$ the formula is Euler's beta integral $\int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \, dt = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$. Selberg proved it in 1944. One proof first treats integer $\gamma$, where the integrand is a polynomial times the beta weights. It then extends to complex $\gamma$ by Carlson's theorem on entire functions of exponential type. Another proof, due to Aomoto, derives a recurrence in $\alpha$ by integrating a derivative and using the symmetry of the integrand. Example: $n = 2$, $\alpha = \beta = 1$, $\gamma = 1$ gives $\int_0^1 \int_0^1 (t_1 - t_2)^2 \, dt_1 dt_2 = \frac{1}{6}$, and the right side is $\frac{\Gamma(1)^2 \Gamma(2)}{\Gamma(3)\Gamma(2)} \cdot \frac{\Gamma(2)^2 \Gamma(3)}{\Gamma(4)\Gamma(2)} = \frac12 \cdot \frac{2}{6} = \frac16$. The bound on $\operatorname{Re} \gamma$ is needed for convergence: the factor $\abs{t_i - t_j}^{2\gamma}$ is not integrable near the diagonal when $\gamma$ is too negative.

Going deeper

For a student

The factor $\prod_{i<j} \abs{t_i - t_j}^{2\gamma}$ is the power of a Vandermonde determinant, the same factor that governs the eigenvalue densities of random matrices. So the Selberg integral computes the normalizing constants of the Jacobi ensembles, and by limits those of the Laguerre and Gaussian ensembles; the Gaussian case is the Mehta integral. It also has $q$-analogues and analogues for other root systems, linked to the Macdonald constant term conjecture.