### 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.

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 - eg Donaldson and Elliott

Residues and Steepest Descents

Riemann-Hilbert Problems - Percy Deift

### Example 1

Simple ,

### Example 2

### Example 3

Trapezoidal Rule - Sinc Stenger http://www.cs.utah.edu/~stenger/

### Example 4

Trapezoidal Rule - Double Exponential Mori

http://mathworld.wolfram.com/DoubleExponentialIntegration.html

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