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Singularities of Integrals : Homology, Hyperfunctions and Microlocal Analysis - Frederic Pham

Singularities of Integrals

Homology, Hyperfunctions and Microlocal Analysis

Paperback Published: 28th April 2011
ISBN: 9780857296023
Number Of Pages: 217

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Bringing together two fundamental texts from Frederic Pham's research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom's isotopy theorems, Frederic Pham's foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis.

Providing a 'must-have' introduction to the singularities of integrals, a number of supplementary references also offer a convenient guide to the subjects covered.

This book will appeal to both mathematicians and physicists with an interest in the area of singularities of integrals.

Frederic Pham, now retired, was Professor at the University of Nice. He has published several educational and research texts. His recent work concerns semi-classical analysis and resurgent functions.

Industry Reviews

From the reviews:

"The book under review is the author's translation of his own recent monograph ... . The book is written in a clear didactic style, almost all topics are illustrated by many examples together with nice pictures, non-formal remarks and comments; it can be recommended to graduate students as well as to mathematicians and physicists who are interested in the studies of singularities of integrals and applications." (Aleksandr G. Aleksandrov, Zentralblatt MATH, Vol. 1223, 2011)

"Singularities of Integrals is a very valuable book in that it deals with serious and important material, and does so in a way accessible to some one wanting to get into the field ... . the target student should really have at least a solid advanced undergraduate background in topology and differential geometry ... and a bit of sheaf theory and hard analysis, to boot. Modulo these prerequisites it's all good stuff." (Michael Berg, The Mathematical Association of America, July, 2011)

Forewordp. ix
Introduction to a topological study of Landau singularities
Introductionp. 3
Differentiable manifoldsp. 7
Definition of a topological manifoldp. 7
Structures on a manifoldp. 7
Submanifoldsp. 10
The tangent space of a differentiable manifoldp. 12
Differential forms on a manifoldp. 17
Partitions of unity on a manifoldp. 20
Orientation of manifolds. Integration on manifoldsp. 22
Appendix on complex analytic setsp. 26
Homology and cohomology of manifoldsp. 29
Chains on a manifold (following de Rham). Stokes' formulap. 29
Homologyp. 31
Cohomologyp. 36
De Rham dualityp. 39
Families of supports. Poincaré's isomorphism and dualityp. 41
Currentsp. 45
Intersection indicesp. 49
Leray's theory of residuesp. 55
Division and derivatives of differential formsp. 55
The residue theorem in the case of a simple polep. 57
The residue theorem in the case of a multiple polep. 61
Composed residuesp. 63
Generalization to relative homologyp. 64
Thom's isotopy theoremp. 67
Ambient isotopyp. 67
Fiber bundlesp. 70
Stratified setsp. 73
Thom's isotopy theoremp. 77
Landau varietiesp. 80
Ramification around Landau varietiesp. 85
Overview of the problemp. 85
Simple pinching. Picard-Lefschetz formulaep. 89
Study of certain singular points of Landau varietiesp. 98
Analyticity of an integral depending on a parameterp. 109
Holomorphy of an integral depending on a parameterp. 109
The singular part of an integral which depends on a parameterp. 114
Ramification of an integral whose integrand is itself ramifiedp. 127
Generalities on covering spacesp. 127
Generalized Picard-Lefschetz formulaep. 130
Appendix on relative homology and families of supportsp. 133
Technical notesp. 137
Sourcesp. 141
Referencesp. 143
Introduction to the study of singular integrals and hyperfunctions
Introductionp. 147
Functions of a complex variable in the Nilsson classp. 149
Functions in the Nilsson classp. 149
Differential equations with regular singular pointsp. 154
Functions in the Nilsson class on a complex analytic manifoldp. 157
Definition of functions in the Nilsson classp. 157
A local study of functions in the Nilsson classp. 159
Analyticity of integrals depending on parametersp. 163
Single-valued integralsp. 163
Multivalued integralsp. 164
An examplep. 167
Sketch of a proof of Nilsson's theoremp. 171
Examples: how to analyze integrals with singular integrandsp. 175
First examplep. 175
Second examplep. 183
Hyperfunctions in one variable, hyperfunctions in the Nilsson classp. 185
Definition of hyperfunctions in one variablep. 185
Differentiation of a hyperfunctionp. 186
The local nature of the notion of a hyperfunctionp. 187
The integral of a hyperfunctionp. 188
Hyperfunctions whose support is reduced to a pointp. 189
Hyperfunctions in the Nilsson classp. 189
Introduction to Sato's microlocal analysisp. 191
Functions analytic at a point x and in a directionp. 191
Functions analytic in a field of directions on Rnp. 191
Boundary values of a function which is analytic in a field of directionsp. 193
The microsingular support of a hyperfunctionp. 196
The microsingular support of an integralp. 197
Construction of the homology sheaf of X over Tp. 201
Homology groups with local coefficientsp. 205
Supplementary referencesp. 207
Indexp. 215
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780857296023
ISBN-10: 0857296027
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 217
Published: 28th April 2011
Publisher: Springer London Ltd
Country of Publication: GB
Dimensions (cm): 22.86 x 15.49  x 1.52
Weight (kg): 0.34

Earn 232 Qantas Points
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