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Theory of Hypergeometric Functions

  • Book
  • © 2011

Overview

Authors:
  1. Kazuhiko Aomoto

    1. Nagoya University, Nagoya, Japan

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  2. Michitake Kita
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  • Reader will understand clearly multidimensional hypergeometric function as a natural extension of the classical one from viewpoint of integrals
  • A quick introduction to rational de Rham cohomology due to A.Grothendieck and P.Deligne and also to holonomic differential equations (or Gauss-Manin connection) and difference equations associated with hypergeometric functions
  • Application of hypergeometric functions to several analytic or geometric problems
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction: the Euler−Gauss Hypergeometric Function

    • Kazuhiko Aomoto, Michitake Kita
    Pages 1-19

  3. Representation of Complex Integrals and Twisted de Rham Cohomologies

    • Kazuhiko Aomoto, Michitake Kita
    Pages 21-101

  4. Arrangement of Hyperplanes and Hypergeometric Functions over Grassmannians

    • Kazuhiko Aomoto, Michitake Kita
    Pages 103-182

  5. Holonomic Difference Equations and Asymptotic Expansion

    • Kazuhiko Aomoto, Michitake Kita
    Pages 183-259

  6. Back Matter

    Pages 261-317
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Keywords

About this book

This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.

Authors and Affiliations

  • Nagoya University, Nagoya, Japan

    Kazuhiko Aomoto

Bibliographic Information

  • Book Title: Theory of Hypergeometric Functions

  • Authors: Kazuhiko Aomoto, Michitake Kita

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-4-431-53938-4

  • Publisher: Springer Tokyo

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer 2011

  • Hardcover ISBN: 978-4-431-53912-4Published: 13 May 2011

  • Softcover ISBN: 978-4-431-54087-8Published: 15 July 2013

  • eBook ISBN: 978-4-431-53938-4Published: 21 May 2011

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XVI, 320

  • Topics: Geometry, Functional Analysis

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