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Transcendental functions --- Calculus

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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.

Algebra -- Congresses. --- Geometry, Algebraic -- Congresses. --- Hypergeometric functions -- Congresses. --- Hypergeometric functions. --- Number theory -- Congresses. --- Hypergeometric functions --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Functions, Hypergeometric --- Mathematics. --- Functional analysis. --- Geometry. --- Functional Analysis. --- Euclid's Elements --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Transcendental functions --- Hypergeometric series

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The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone decreasing sequence of non-negative numbers with a view to the classification of operators according to the way in which these numbers approach a limit: approximation numbers provide an especially important example of such numbers. The asymptotic behavior of the s-numbers of Hardy operators acting between Lebesgue spaces is determined here in a wide variety of cases. The proof methods involve the geometry of Banach spaces and generalized trigonometric functions; there are connections with the theory of the p-Laplacian.

Eigenvalues --- Embeddings (Mathematics) --- Trigonometrical functions --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Algebra --- Applied Mathematics --- Trigonometrical functions. --- Circular functions --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Approximation theory. --- Functional analysis. --- Differential equations. --- Special functions. --- Analysis. --- Approximations and Expansions. --- Functional Analysis. --- Special Functions. --- Ordinary Differential Equations. --- Mathematics Education. --- Study and teaching. --- Special functions --- Mathematical analysis --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- 517.1 Mathematical analysis --- Math --- Science --- Transcendental functions --- Global analysis (Mathematics). --- Functions, special. --- Differential Equations. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematics—Study and teaching .