Introduction to complex theory of differential equations.

*(English)*Zbl 1384.58002
Frontiers in Mathematics. Basel: Birkhäuser/Springer (ISBN 978-3-319-51743-8/pbk; 978-3-319-51744-5/ebook). x, 138 p. (2017).

The aim of this book is to explain the theory of complex differential equations on a complex manifold.

After introducing quickly Leray’s residue theory on a complex manifold and the notion of a ramified integral (a multivalued holomorphic function given by the integral of a closed holomorphic form on a moving homological compact cycle) the authors give Leray’s theorem on the asymptotics of such an integral near a simple pinch point.

In Chapters 4 and 5, they introduce the ramified Fourier transform which will be used in the next chapters to study the Cauchy problem with constant coefficients and the singularities of the the solutions of this problem. Then, they study in Chapter 8 the Cauchy problem with variable coefficients and give Leray’s uniformization theorem.

The last two chapters are devoted to applications: the balayage inwards problem and the mother body problem.

In this book, many simple examples are described in details, and this help the reader to understand the general statements and the way to use them in concrete situations.

After introducing quickly Leray’s residue theory on a complex manifold and the notion of a ramified integral (a multivalued holomorphic function given by the integral of a closed holomorphic form on a moving homological compact cycle) the authors give Leray’s theorem on the asymptotics of such an integral near a simple pinch point.

In Chapters 4 and 5, they introduce the ramified Fourier transform which will be used in the next chapters to study the Cauchy problem with constant coefficients and the singularities of the the solutions of this problem. Then, they study in Chapter 8 the Cauchy problem with variable coefficients and give Leray’s uniformization theorem.

The last two chapters are devoted to applications: the balayage inwards problem and the mother body problem.

In this book, many simple examples are described in details, and this help the reader to understand the general statements and the way to use them in concrete situations.

Reviewer: Daniel Barlet (Vandœuvre-les-Nancy)

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J32 | Boundary value problems on manifolds |

58J37 | Perturbations of PDEs on manifolds; asymptotics |

58J47 | Propagation of singularities; initial value problems on manifolds |

58Z05 | Applications of global analysis to the sciences |

35A20 | Analyticity in context of PDEs |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32W50 | Other partial differential equations of complex analysis in several variables |

35R01 | PDEs on manifolds |