Let $M$ be a complex manifold of dimension $n$, and $D$ be a smooth hypersurface. Let $\varphi$ be a $C^{\infty}$ $ k$-form on $M \backslash D$. We say that $\varphi$ has logarithmic singularities on $D$ if for any point $m_{0} \in D$, there exists a neighborhood $U$ of $m_{0}$ in $M$, where $D$ is defined by a local equation $s=0$ : $$ D \cap U=\{m \in U \mid s(m)=0\} $$ and the form $\varphi$ can be written in $U \backslash D$ as $$\varphi=\frac{d s}{s} \wedge \omega+\theta,$$ where $\omega$ and $\theta$ are $C^{\infty}$ forms on $U$ of degree $k-1$ and $k$ respectively.
In p.104 of the book Ancona--Gaveau, Differential forms on singular varieties. De Rham and Hodge theory simplified, the Lemma 6.2 reads: Res commutes with $d: \operatorname{Res} d \varphi=d \operatorname{Res} \varphi$.
Here is the proof of Lemma 6.2:
(ii) We write $\varphi$ as $\varphi=\frac{d s}{s} \wedge \omega+\theta$. Then $$ d \varphi=\frac{d s}{s} \wedge d \omega+d \theta. $$ Hence $\operatorname{Res} d \varphi=\left.d \omega\right|_{D}=d \operatorname{Res} \varphi$.
My opinion: According to the above argument, this should be reduced to show $$d(ds/s)=0$$ (in the current sense; outside $D$ this is of course right. But over the whole of $U$ it is a current calculation). One can easily check that $d(ds/s)=0$ is equivalent to $\frac{\partial}{\partial \bar{z}}( \frac{1}{z})=0$. However I'm pretty sure that $$\frac{\partial}{\partial \bar{z}}( \frac{1}{z})=\pi \delta_{0}$$ via the generalized Cauchy integral formula, here we view $s$ as the coordinate function $z_1$ and the Dirac measure $\delta_0$ is w.r.t. the $z_1$ plane.
Question: Has anyone seen this result that Res commutes with $d$ in any other book or paper?(I think this result may be wrong)
Any clues is appreciated!