Gauss–Jacobi quadrature can be used to approximate integrals of the form where ƒ is a smooth function on [−1, 1] and α, β > −1.
Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points.
Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes α = β = −0.5 (+0.5).
Gauss–Jacobi quadrature uses ω(x) = (1 − x)α (1 + x)β as the weight function.
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. The error term (difference between approximate and accurate value) is: where