Many quadrature rules are derived from others by first making a change-of-variable in the integral and then applying an existing quadrature rule.
Quadrature error can be represented by a contour integral over a curve enclosing the interval. This error can be estimated using residues around poles, integrals along branch cuts, or saddle points
My idea is to unify the existing cases by defining a Riemann surface by the change-of-variable and then the error is represented by an integral over the Riemann surface.
Integral Representation of Error Donaldson and Elliott
Residues and Steepest Descents
Trapezoidal Rule - Sinc, Double Exponential Stenger, Mori
Rational Basis Functions Boyd
Elliott and Johnson Gauss Legendre