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This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. It shows that hypergeometric integrals generally satisfy holonomic system of linear differential equations.

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1 Introduction: the Euler-Gauss Hypergeometric Function.- 2 Representation of Complex Integrals and Twisted de Rham Cohomologies.- 3 Hypergeometric functions over Grassmannians.- 4 Holonomic Difference Equations and Asymptotic Expansion References Index.

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