where ƒ is a smooth function on [−1, 1] and α, β > −1. The interval [−1, 1] can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with α = β = 0. Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes α = β = -0.5 (+0.5). More generally, the special case α = β turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.
Gauss–Jacobi quadrature uses ω(x) = (1 − x)α (1 + x)β as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on n points has the form
where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula
where Γ denotes the Gamma function and P(α, β)
n(x) the Jacobi polynomial of degree n.
- Rabinowitz, Philip (2001), "§4.8-1: Gauss–Jacobi quadrature", A First Course in Numerical Analysis (2nd ed.), New York: Dover Publications, ISBN 978-0-486-41454-6.