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
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从局部展开式解析函数的全局重构以及将和转换为积分的新通用方法
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
X. Xia
中科院分区：
文献类型：
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作者：
O. Costin;X. Xia
提出了一种新的求和方法，将一类比较广泛的泰勒级数和无穷和问题转化为积分问题。全局行为，如解析延拓，位置的奇点，渐近大值的变量和渐近位置的零，从而遵循，通过积分表示，从泰勒系数在一个点，说零。该方法在某种意义上可以看作是柯西公式的逆。它可以在一个或多个复杂变量中工作。重构函数的全局解析结构与系数作为其指数的函数的性质之间存在对偶性。一类发散级数的Borel可和性作为一个副产品。
A new summation method is introduced to convert a relatively wide family of 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 coefficients as a function of their index. Borel summability of a class of divergent series follow as a byproduct.