On principal value and standard extension of distributions

For a holomorphic function $f$ on complex manifold $\mathscr {M}$ we explain in this article that the distribution associated to $\lvert f\rvert^{2\alpha } (\textrm{Log} \lvert f\rvert^2)^q f^{-N}$ by taking corresponding limit sets $\{ f\rvert \geq \varepsilon \}$ when $\varepsilon $ goes $0$, coincides for $\Re (\alpha ) non negative and $q, N \in \mathbb {N}$, with value at $\lambda = \alpha of meromorphic extension f\rvert^{2\lambda f\rvert^2)^qf^{-N}$. This implies any $\mathcal {D}_{\mathscr {M}}$-module generated such has standard property. {O}_\mathscr torsion result distribution. As an application determine generators conjugate modules regular holonomic {D}$-modules introduced studied [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] operators which annihilate roots universal equation degree $k$, arXiv:2101.01895] $z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} 0$.