Differentiable manifolds -- Homology and cohomology of manifolds -- Leray’s theory of residues -- Thom’s isotopy theorem -- Ramification around Landau varieties -- Analyticity of an integral depending on a parameter -- Ramification of an integral whose integrand is itself ramified -- Functions of a complex variable in the Nilsson class -- Functions in the Nilsson class on a complex analytic manifold -- Analyticity of integrals depending on parameters -- Sketch of a proof of Nilsson’s theorem -- Examples: how to analyze integrals with singular integrands -- Hyperfunctions in one variable, hyperfunctions in the Nilsson class -- Introduction to Sato’s microlocal analysis.

Bringing together two fundamental texts from Frédéric 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, Frédéric 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. Frédéric 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.