Gauss–Jacobi quadrature

Remarks

The proof is the general argument for Gaussian quadrature. Divide $p$ by $P_n^{(\alpha,\beta)}$ and use the orthogonality of $P_n^{(\alpha,\beta)}$ to all polynomials of lower degree for the weight $\omega$. Then integrate the remainder, of degree at most $n - 1$, by interpolation. Applying the exactness to $\ell_i^2$, of degree $2n - 2$, gives $w_i = \int \omega \ell_i^2 > 0$. Example: $\alpha = \beta = 0$ gives Gauss–Legendre quadrature. Example: $\alpha = \beta = -\tfrac12$ gives Gauss–Chebyshev quadrature, with nodes $x_i = \cos\frac{(2i - 1)\pi}{2n}$ and equal weights $w_i = \pi / n$; for $n = 1$ the rule is $\pi f(0)$, and indeed $\int_{-1}^1 (1 - x^2)^{-1/2} dx = \pi$. The condition $\alpha, \beta > -1$ is needed for $\omega$ to be integrable; for $\alpha \le -1$ even $\int \omega$ diverges.

Going deeper

For a student

The family contains the Gauss–Legendre and Gauss–Chebyshev rules as special cases. It is used for integrands with known algebraic endpoint singularities, for example in boundary integral methods and in spectral methods based on Jacobi polynomials. The nodes and weights are computed in practice from the three-term recurrence of the Jacobi polynomials, as eigenvalues and eigenvector components of a symmetric tridiagonal matrix.