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Integrals and differential equations

  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

In this chapter we shall prove that many-valued functions, given as integrals of a holomorphic differential form over classes of continuous families of homologies, vanishing at the critical point of a holomorphic function, are all solutions of an ordinary homogeneous linear differential equation, the order of which is not greater than the multiplicity of the critical point The analysis of this phenomenon leads to the concept of the Gauss-Manin connection in the fibration of vanishing cohomologies associated with the Milnor fibration of the critical point.

Keywords

Holomorphic Form Versal Deformation Mixed Hodge Structure Regular Singular Point Monodromy Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • V. I. Arnold
  • S. M. Gusein-Zade
    • 1
  • A. N. Varchenko
    • 2
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsThe University of North Carolina at Chapel HillChapel HillUSA

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