Ebook: Differential Equations on Complex Manifolds

The present monograph is devoted to the complex theory of differential equations. Not yet a handbook, neither a simple collection of articles, the book is a first attempt to present a more or less detailed exposition of a young but promising branch of mathematics, that is, the complex theory of partial differential equations. Let us try to describe the framework of this theory. First, simple examples show that solutions of differential equations are, as a rule, ramifying analytic functions. and, hence, are not regular near points of their ramification. Second, bearing in mind these important properties of solutions, we shall try to describe the method solving our problem. Surely, one has first to consider differential equations with constant coefficients. The apparatus solving such problems is well-known in the real the­ ory of differential equations: this is the Fourier transformation. Un­ fortunately, such a transformation had not yet been constructed for complex-analytic functions and the authors had to construct by them­ selves. This transformation is, of course, the key notion of the whole theory.

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This volume contains a unique, systematic presentation of the general theory of differential equations on complex manifolds.
The six chapters deal with questions concerning qualitative (asymptotic) theory of partial differential equations as well as questions about the existence of solutions in spaces of ramifying functions. Furthermore, much attention is given to applications. In particular, important problems connected with the continuation of (real) solutions to differential equations and with mathematical theory of diffraction are solved here.
The book is self-contained, and includes up-to-date results. All necessary terminology is explained.
For graduate students and researchers interested in differential equations in partial derivatives, complex analysis, symplectic and contact geometry, integral transformations and operational calculus, and mathematical physics.

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This volume contains a unique, systematic presentation of the general theory of differential equations on complex manifolds.
The six chapters deal with questions concerning qualitative (asymptotic) theory of partial differential equations as well as questions about the existence of solutions in spaces of ramifying functions. Furthermore, much attention is given to applications. In particular, important problems connected with the continuation of (real) solutions to differential equations and with mathematical theory of diffraction are solved here.
The book is self-contained, and includes up-to-date results. All necessary terminology is explained.
For graduate students and researchers interested in differential equations in partial derivatives, complex analysis, symplectic and contact geometry, integral transformations and operational calculus, and mathematical physics.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-40
Some Questions of Analysis and Geometry of Complex Manifolds....Pages 41-136
Symplectic and Contact Structures....Pages 137-193
Integral Transformations of Ramified Analytic Functions....Pages 195-249
Laplace-Radon Integral Operators....Pages 251-287
Cauchy Problem in Spaces of Ramified Functions....Pages 289-404
Continuation of Solutions to Elliptic Equations....Pages 405-474
Back Matter....Pages 475-508

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This volume contains a unique, systematic presentation of the general theory of differential equations on complex manifolds.
The six chapters deal with questions concerning qualitative (asymptotic) theory of partial differential equations as well as questions about the existence of solutions in spaces of ramifying functions. Furthermore, much attention is given to applications. In particular, important problems connected with the continuation of (real) solutions to differential equations and with mathematical theory of diffraction are solved here.
The book is self-contained, and includes up-to-date results. All necessary terminology is explained.
For graduate students and researchers interested in differential equations in partial derivatives, complex analysis, symplectic and contact geometry, integral transformations and operational calculus, and mathematical physics.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-40
Some Questions of Analysis and Geometry of Complex Manifolds....Pages 41-136
Symplectic and Contact Structures....Pages 137-193
Integral Transformations of Ramified Analytic Functions....Pages 195-249
Laplace-Radon Integral Operators....Pages 251-287
Cauchy Problem in Spaces of Ramified Functions....Pages 289-404
Continuation of Solutions to Elliptic Equations....Pages 405-474
Back Matter....Pages 475-508
....