Quadrature formulas from rational approximations

It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new method to derive all kinds of quadrature formulas as well as fundamental insight into the mathematics of quadrature. Intervals or curves of quadrature nodes correspond to near-optimal branch cuts of the Cauchy transform.