series $\CC[[\vep,\vep^{-1}]$.

An element of $\CC[[\vep,\vep^{-1}] [T]$ is of the form $\sum_{n=0}^M a_n(\vep)T^n$ with $$a_n(\vep)=\sum_{k\geq N_n} a_{n,k}\vep^k \in \CC[[\vep,\vep^{-1}], 0\leq n\leq M.$$ Taking $N=\min_{0\leq n\leq M} N_n$ and letting $a_{n,k}=0$ for $N\leq k< N_n$, we have $$\sum_{n=0}^M a_n(\vep)T^n = \sum_{n=0}^M \big(\sum_{k\geq N} a_{n,k}\vep^k\big) T^n =\sum_{k\geq
N} \big(\sum_{n=0}^M a_{n,k} T^n\big) \vep^k.$$ This gives an element of the [**algebra of log Laurent series**]{} $\CC[T][[\vep,\vep^{-1}]$ with coefficients in $\CC[T]$. Combining with Eq. (), we obtain a natural algebra injection $$u: \CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]  \to \CC[T][[\vep,\vep^{-1}]
\mlabel{eq:formulau}$$ with which we identify $\CC\{\{\vep,\vep^{-1}\} [\ln (-\vep)]$ as a subalgebra of $\CC[T][[\vep,\vep^{-1}]$.

For any $\vec s\in \ZZ ^k$, $\vec r \in {\mathbb}{Z}_{>0}^k$, $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC \{\{\vep,\vep^{-1}\}[ln(-\vep)]$ and can thus be regarded as an element in $\CC[T][[\vep,\vep^{-1}]$ by Eq. (). If $\vec s$ is in ${\mathbb}{Z}_{\le 0}^k$, then $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC\{\{\vep,\vep^{-1}\}$.

First notice that, for $r,i\in \ZZ_{>0}$, $$\sum _{n\geq i}e^{nr\vep }=\frac {1}{1-e^{r\vep}}\,e^{ir\vep}
 \mlabel{eq:geo}$$ has a Laurent series expansion at $\vep=0$. Since $Z(\wvec{s}{r};\vep)$ is uniformly convergent on compact subsets in $Re(\vep)<0$, by repeatedly differentiating Eq. (), we see that, for $s\in \ZZ_{<0}$, $$\begin{aligned}
\sum _{n\geq i}n^{-s}e^{nr\vep}
&=& r^s\Big( \sum_{p=0}^{-s} \binc{-s}{p} \big(\frac{1}{1-e^{r\vep}}\big)^{(p)}(e^{ir\vep})^{(-s-p)}\Big) \notag\\
&=&\sum _{p=0}^{-s}\binc{-s}{p}
\big(\frac{1}{1-e^{r\vep}}\big)^{(p)}
\frac{e^{ir\vep}}{r^{p}\,i^{s+p}}
\mlabel{eq:diffsum}\end{aligned}$$ has a Laurent series expansion at 