 Order of a meromorphic function along a hypersurface

At a generic smooth point of an irreducible complex analytic hypersurface $Y$, a nonzero meromorphic function has the form $f=t^{m}u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The unique factorization domain property of the local ring of a complex manifold makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the principal divisor on a complex manifold of $f$.