Multidimensional residues and their applications. Transl. from the Russian by E. J. F. Primrose. Transl. edited by S. Gelfand.

*(English)*Zbl 0758.32001
Translations of Mathematical Monographs. 103. Providence, RI: American Mathematical Society (AMS). x, 188 p. (1992).

There are several approaches to multidimensional residues. The book presents a systematic account of residues connected with integrals of meromorphic and semimeromorphic forms.

Chapter I contains some preliminary information from the theory of analytic sets and algebraic topology, and is more for ease of references than for study.

Chapter II is devoted to a detailed study of residues associated with holomorphic mappings that preserve the dimension (local residues). It gives an integral definition of a local residue, its basic properties and an algebraic interpretation in terms of the trace with respect to some extension of the field of meromorphic functions. Theorem on the total sum of residues and a compact manifold is proved, and the case of a polynomial mapping is studied in detail.

Chapter III deals with residues associated with holomorphic mappings that lower the dimension (residual currents). The existence of residual currents is proved, and their properties are studied.

In Chapter IV residues are applied to function theory and algebraic geometric: integral representations of holomorphic functions, investigation of holomorphic functions on analytic sets, calculating the multiplicity of a mapping, ideals in a polynomial ring.

Chapter V is devoted to applications of residues to double series and integrals.

The book will be useful to researchers in complex analysis, and is acceptible for graduate students.

Chapter I contains some preliminary information from the theory of analytic sets and algebraic topology, and is more for ease of references than for study.

Chapter II is devoted to a detailed study of residues associated with holomorphic mappings that preserve the dimension (local residues). It gives an integral definition of a local residue, its basic properties and an algebraic interpretation in terms of the trace with respect to some extension of the field of meromorphic functions. Theorem on the total sum of residues and a compact manifold is proved, and the case of a polynomial mapping is studied in detail.

Chapter III deals with residues associated with holomorphic mappings that lower the dimension (residual currents). The existence of residual currents is proved, and their properties are studied.

In Chapter IV residues are applied to function theory and algebraic geometric: integral representations of holomorphic functions, investigation of holomorphic functions on analytic sets, calculating the multiplicity of a mapping, ideals in a polynomial ring.

Chapter V is devoted to applications of residues to double series and integrals.

The book will be useful to researchers in complex analysis, and is acceptible for graduate students.

Reviewer: A.Yu.Rashkovsky (Khar’kov)

##### MSC:

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

32A27 | Residues for several complex variables |

32C30 | Integration on analytic sets and spaces, currents |