of Oxford) introduces the constructive approximation of polynomials and rational functions, and extends the techniques to interpolation, quadrature, rootfinding,

analytic continuation, extrapolation of sequence and series, and solution of differential equations.

The results of the present paper are related to the approximation of functions which are continuous on [0,1] and possess an

analytic continuation into a disk {z : [absolute value of z] < a}, a > 0, by their q-Bernstein polynomials in the case q > 1.

1), and the final result follows by an appeal to the principle of

analytic continuation.

Keywords Hexagon-number, Abel's summation formula, the

analytic continuation.

Emphasizing how complex analysis is a natural outgrowth of multivariable real calculus, this graduate textbook introduces the Cauchy integral formula, the properties and behavior of holomorphic functions, harmonic functions,

analytic continuation, topology, Mergelyan's theorem, Hilbert spaces, and the prime number theorem.

It can be shown by

analytic continuation that when Im([omega]) < 0, the algebraic function [[xi].

He also includes integral solutions and

analytic continuation to show how all the methods are related.

For instance, it contains the inverses of the Askey-Wilson divided difference operators [20] and the q-Fourier transform [8] as special cases after certain

analytic continuation with respect to the free parameter.

Analytic continuation and special values are given by the contour integral representation of [[zeta].

They begin by describing complex numbers and their elementary properties, including power series, powers and logarithms and the geometric properties of simple functions, analytic functions such as differentiation and integration in the complex frame, Cauchy's integral formula, Taylor and Laurent series and

analytic continuation, contour integration, conformal mapping, including the Joukowsky and Schwarz-Christoffel transformations, special functions such as the gamma function and the Lefendre and Bessel functions, asymptotic methods such as that of Laplace, transform methods such as Fourier transforms, and special techniques such as the Weiner-Hopf methods, the kernel decomposition and using approximate kernels.

The 19 papers in the proceedings discuss such topics as quantum field theory and the volume conjecture, representations and the colored Jones polynomial of a torus knot, delta-groupoids and ideal triangulations, spin foam state sums and the Chern-Simons theory, Yang-Mills in two dimensions and Chern-Simons in three, fermionization and convergent perturbation expansions in Chern-Simons gauge theory, and

analytic continuation of Chern-Simons theory.

This integral may be viewed as the

analytic continuation of the [sub.