%
\label{def:laurent-expansion}
Let $f = \{f_k\}_{k \geq 1}$
be an $\omega$-germ transformer
on~$\tau^3$,
with bipolar decomposition
$f = e_+ f_+ + e_- f_-$.
The \textbf{Laurent expansion} of~$f$
at a $\tau$-admissible point~$x$
is the formal double series:
\[
    \boxed{%
    f
    \;=\;
    \sum_{n \in \mathbb{Z}}
    a_n^{(+)}\, e_+\, \phi_n^{(+)}
    \;\;+\;\;
    \sum_{n \in \mathbb{Z}}
    a_n^{(-)}\, e_-\, \phi_n^{(-)},}
\]
where the \textbf{spectral coefficients}
$a_n^{(\pm)} \in \mathbb{R}$
are determined by:
\[
    a_n^{(+)}
    \;=\;
    \lim_{k \to \infty}\;
    \frac{1}{P_k}
    \sum_{a=0}^{P_k - 1}
    f_+^{(k)}(a)\,
    \overline{\chi_n^{(\gamma)}(a)},
    \qquad
    a_n^{(-)}
    \;=\;
    \lim_{k \to \infty}\;
    \frac{1}{P_k}
    \sum_{a=0}^{P_k - 1}
    f_-^{(k)}(a)\,
    \overline{\chi_n^{(\eta)}(a)}.
\]
The sums are \textbf{finite discrete Fourier transforms}
at each stage~$k$,
and the limits exist by tower coherence.
The index $n$ ranges over $\mathbb{Z}$:
positive $n$ corresponds to
\textbf{regular terms}
(present in holomorphic functions),
while negative $n$ corresponds to
\textbf{principal part terms}
(present only at singularities).