Global reconstruction of analytic functions from local expansions and a new general method of converting sums into integrals

Details Global reconstruction of analytic functions from local expansions and a new general method of converting sums into integrals Global reconstruction of analytic functions from local expansions and a new general method of converting sums into integrals

2006-12-05

arxiv.org/abs/math/0612121v1

Mathematics

Complex Variables

Scientific paper

A new summation method is introduced to convert a relatively wide family of local Taylor series and infinite sums into integrals. Global behavior such as analytic continuation, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros thereby follow, through the integral representations, from the Taylor coefficients at a point, say zero. The method can be viewed in some sense as the inverse of Cauchy's formula. It can work in one or several complex variables. There is a duality between the global analytic structure of the reconstructed function and the properties of the coefficient as a function of their index. Borel summability of a class of divergent series follow as a biproduct.

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