Asymptotics of ramified integrals

In the previous chapter, we saw that the singularities of ramified integrals lie on Landau manifolds. So the question then arises about a more precise description of these singularities. The aim of this chapter is to answer this question. In more detail, we study ramified integrals near generic points of Landau manifolds. Namely, we show that the ramification of the homology class, over which we integrate, is described by Picard-Lefschetz formulas and that the asymptotics of the integral is given by Leray’s formulas. Slightly simplifying the situation, the main result can be formulated as follows: generically, near regular points of the Landau manifold, ramified integrals have singularities of one of the three types: square root singularity, logarithmic singularity, or pole. © Springer International Publishing AG 2017.