### 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

Chebyshev ### 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

### Example 6

Rational Basis Functions

Boyd

http://portal.acm.org/citation.cfm?id=33099

### Example 7

Elliott and Johnson Gauss Legendre

http://anziamj.austms.org.au/V40/E006/home.html