### Quick description

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.

### Prerequisites

==Riemann Surfaces==

Integral Representation of Error Donaldson and Elliott

Residues and Steepest Descents

Riemann-Hilbert Problems

Percy Deift

### Example 1

,

### Example 2

Chebyshev

### Example 3

Trapezoidal Rule - Sinc, Double Exponential

Stenger, Mori

### Example 4

Sigmoidal

Elliott

### Example 5

Rational Basis Functions

Boyd

### Example 6

Elliott and Johnson

Gauss Legendre

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