  We investigate the analytic structure of functions defined by integrals with
integrands singular on a finite union of quadrics. The main motivation comes
from Feynman integrals which belong to this class. Using isotopy techniques we
derive the Landau equations in momentum space from the theory of Feynman
integrals and generalize these equations to naturally include singularities of
the second type. For this purpose we introduce a regularization of analytic
families of quadratic forms rendering the isotopy techniques applicable. In the
case of Feynman integrals we comment on what is known about the behavior on the
principal branch where only specific solutions of the Landau equations
contribute to non-analytic points. Finally we discuss compatibility with
renormalization.
