is also closed under the indefinite integral operator: the antiderivatives of any $f\in \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ are in $\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$.

Let $f(\vep)\in \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$. Then $$f(\vep)=\sum_{n=0}^M a_n(\vep)\ln^n (- \vep)
=\sum_{n=0}^M \big(\sum_{k\geq N_n} a_{n,k}\vep^k \ln^n(-\vep)\big)$$ with $\sum_{k\geq N_n} a_{n,k}\vep^k \in \CC\{\{\vep,\vep^{-1}\}$. For each $0\leq n\leq M$, the series for the inside sum converges absolutely and uniformly in a nonempty open interval of $\{\vep\in \CC\,\big |\, -\infty<\vep <0\}$. Thus the series can be differentiated and integrated term by term. Thus we only need to show that the derivative and anti-derivatives of $\vep^k \ln ^n (-\vep)$, $k\in \ZZ, n\in \ZZ_{\geq 0}$ are linear combinations of functions of the same form. This is easy to check for derivatives.

For anti-derivatives, we use induction on $n$. It is clear when $n=0$. The induction step follows from the integration by parts formula $$\int \vep^k \ln ^n (-\vep) d\vep = \frac{1}{k+1}\vep^{k+1} \ln ^n (-\vep)
+ \frac{n}{k+1} \int  \vep^{k} \ln ^{n-1} (-\vep) d\vep$$ when $k\neq -1$ and $ \int \frac{\ln ^n (-\vep)}{\vep}d\vep = \frac{1}{n+1} \ln ^{n+1} (-\vep)+C.$

Because of Lemma , we have $$\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]\cong
\CC\{\{\vep,\vep^{-1}\} [T]
\hookrightarrow \CC[[\vep,\vep^{-1}][T]
\mlabel{eq:lau2}$$ sending $-\ln (-\vep)$ to $T$. Here $\CC[[\vep,\vep^{-1}][T]$ denotes the polynomial algebra over the formal Laurent 