Why Gaussian quadrature in the complex plane?

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This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude c T A −1 b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c T x (k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/λ. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.