Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are closely related to the localization and the local cohomology of $M$. We show that the $b$-function, if it exists, controls these modules and present general algorithms for computing these modules and the $b$-function if it exists without any further assumptions. We also give some examples of multiplicity computation of such $D$-modules including a possibly well-known explicit formula for the localization of the polynomial ring by a hyperplane arrangement.
Digital Object Identifier: 10.2969/aspm/07710353