| Foreword | p. ix |
| Introduction to a topological study of Landau singularities | |
| Introduction | p. 3 |
| Differentiable manifolds | p. 7 |
| Definition of a topological manifold | p. 7 |
| Structures on a manifold | p. 7 |
| Submanifolds | p. 10 |
| The tangent space of a differentiable manifold | p. 12 |
| Differential forms on a manifold | p. 17 |
| Partitions of unity on a ∞ manifold | p. 20 |
| Orientation of manifolds. Integration on manifolds | p. 22 |
| Appendix on complex analytic sets | p. 26 |
| Homology and cohomology of manifolds | p. 29 |
| Chains on a manifold (following de Rham). Stokes' formula | p. 29 |
| Homology | p. 31 |
| Cohomology | p. 36 |
| De Rham duality | p. 39 |
| Families of supports. Poincaré's isomorphism and duality | p. 41 |
| Currents | p. 45 |
| Intersection indices | p. 49 |
| Leray's theory of residues | p. 55 |
| Division and derivatives of differential forms | p. 55 |
| The residue theorem in the case of a simple pole | p. 57 |
| The residue theorem in the case of a multiple pole | p. 61 |
| Composed residues | p. 63 |
| Generalization to relative homology | p. 64 |
| Thom's isotopy theorem | p. 67 |
| Ambient isotopy | p. 67 |
| Fiber bundles | p. 70 |
| Stratified sets | p. 73 |
| Thom's isotopy theorem | p. 77 |
| Landau varieties | p. 80 |
| Ramification around Landau varieties | p. 85 |
| Overview of the problem | p. 85 |
| Simple pinching. Picard-Lefschetz formulae | p. 89 |
| Study of certain singular points of Landau varieties | p. 98 |
| Analyticity of an integral depending on a parameter | p. 109 |
| Holomorphy of an integral depending on a parameter | p. 109 |
| The singular part of an integral which depends on a parameter | p. 114 |
| Ramification of an integral whose integrand is itself ramified | p. 127 |
| Generalities on covering spaces | p. 127 |
| Generalized Picard-Lefschetz formulae | p. 130 |
| Appendix on relative homology and families of supports | p. 133 |
| Technical notes | p. 137 |
| Sources | p. 141 |
| References | p. 143 |
| Introduction to the study of singular integrals and hyperfunctions | |
| Introduction | p. 147 |
| Functions of a complex variable in the Nilsson class | p. 149 |
| Functions in the Nilsson class | p. 149 |
| Differential equations with regular singular points | p. 154 |
| Functions in the Nilsson class on a complex analytic manifold | p. 157 |
| Definition of functions in the Nilsson class | p. 157 |
| A local study of functions in the Nilsson class | p. 159 |
| Analyticity of integrals depending on parameters | p. 163 |
| Single-valued integrals | p. 163 |
| Multivalued integrals | p. 164 |
| An example | p. 167 |
| Sketch of a proof of Nilsson's theorem | p. 171 |
| Examples: how to analyze integrals with singular integrands | p. 175 |
| First example | p. 175 |
| Second example | p. 183 |
| Hyperfunctions in one variable, hyperfunctions in the Nilsson class | p. 185 |
| Definition of hyperfunctions in one variable | p. 185 |
| Differentiation of a hyperfunction | p. 186 |
| The local nature of the notion of a hyperfunction | p. 187 |
| The integral of a hyperfunction | p. 188 |
| Hyperfunctions whose support is reduced to a point | p. 189 |
| Hyperfunctions in the Nilsson class | p. 189 |
| Introduction to Sato's microlocal analysis | p. 191 |
| Functions analytic at a point x and in a direction | p. 191 |
| Functions analytic in a field of directions on Rn | p. 191 |
| Boundary values of a function which is analytic in a field of directions | p. 193 |
| The microsingular support of a hyperfunction | p. 196 |
| The microsingular support of an integral | p. 197 |
| Construction of the homology sheaf of X over T | p. 201 |
| Homology groups with local coefficients | p. 205 |
| Supplementary references | p. 207 |
| Index | p. 215 |
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