Differential Equations on Complex Manifolds

1. Front Matter

   Pages i-xii

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2. Introduction

   • Boris Sternin, Victor Shatalov

   Pages 1-40

3. Some Questions of Analysis and Geometry of Complex Manifolds

   • Boris Sternin, Victor Shatalov

   Pages 41-136

4. Symplectic and Contact Structures

   • Boris Sternin, Victor Shatalov

   Pages 137-193

5. Integral Transformations of Ramified Analytic Functions

   • Boris Sternin, Victor Shatalov

   Pages 195-249

6. Laplace-Radon Integral Operators

   • Boris Sternin, Victor Shatalov

   Pages 251-287

7. Cauchy Problem in Spaces of Ramified Functions

   • Boris Sternin, Victor Shatalov

   Pages 289-404

8. Continuation of Solutions to Elliptic Equations

   • Boris Sternin, Victor Shatalov

   Pages 405-474

9. Back Matter

   Pages 475-508

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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.

• Complex analysis
• contact geometry
• integral transform
• manifold
• operational calculus
• partial differential equation
• partial differential equations

• Moscow State University, Moscow, Russia

  Boris Sternin

• Moscow Institute of Electronics and Mathematics, Moscow, Russia

  Victor Shatalov

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