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.

I-Q_n = \frac{1}{2\pi i} \int_C \G(w) f(g(w)) g'(w) dw [\maths] where encloses and has poles at the quadrature 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.