and is supposed to have at 0 a regular point or a simple pole, so actually $a_n = 0$ for all $n \ge 0$ and:

$$ \zeta(z) = \frac{a_{-2}}{z^2} + \frac{a_{-1}}{z} $$

with $a_{-2} \neq 0$ and $a_{-1}$ is the opposite of the double pole residue res (see Section 2.1). By a linear change of variable on $z$ we may assume that $a_{-2} = 1$:

$$ \zeta(z) = \frac{1}{z^2} - \frac{\mathrm{res}}{z}. $$

We immediately deduce that:

**Lemma.** For each value of the residue, there is one and only one local isomorphism class of double poles with this residue realized in the class C.

We saw in Section 3.1.2 that there are exactly two possible classes of double poles for each value of the residue, one centered and one non-centered. This already shows that not all double pole local isomorphism classes can be realized in class C.

We will figure out with purely analytical methods which of the two classes is realized. We will see that it is not easy, outside some special cases.

The developing map of any germ of affine chart is a map $\phi : \tilde{\mathbb{C}}^* \to \mathbb{C}$ that satisfies $\phi''/\phi' = \zeta = \frac{1}{z^2} - \frac{\mathrm{res}}{z}$. So $\log \phi' = -\frac{1}{z} - \mathrm{res} \times \log z$ up to an additive constant and

$$ \phi' = z^{-\mathrm{res}} \exp(-1/z) $$

up to a multiplicative constant.

When $\mathrm{res} = m \in \mathbb{Z}$, we are in a favorable situation: $\phi'$ is well-defined on $\mathbb{C}^*$, has a classical Laurent power series expansion, and $\phi$ is a logarithm plus a Laurent power series:

$$ \phi' = \sum_{n \ge 0} \frac{(-1)^n}{n!} z^{-m-n} $$

up to a multiplicative constant so, up to post-composition with an affine bijection of $\mathbb{C}$:
A. If $m \le 1$, write $m = 1 - n_0$ and we have

$$ \phi = \frac{(-1)^{n_0}}{n_0!} \log z + \sum_{n \ge 0, n \ne n_0} \frac{(-1)^n}{n!(-m-n+1)} z^{-m-n+1}. $$

B. If $m \ge 2$:

$$ \phi = \sum_{n \ge 0} \frac{(-1)^n}{n!(-m-n+1)} z^{-m-n+1}. $$

In case A the affine holonomy of $\phi$ is a non-zero translation and the double-pole is non-centered. In case B is the identity and the double pole is centered.

When $-\mathrm{res} \notin \mathbb{Z}$, things are more complicated. The map $\phi$ can still be computed as a generalized Laurent power series (without log term):

$$ \phi = \sum_{n \in \mathbb{N}} \frac{(-1)^n}{n!(-\mathrm{res} - n + 1)} z^{-\mathrm{res} - n + 1} = z^{-\mathrm{res} + 1} \sum_{n \in \mathbb{N}} \frac{(-1)^n}{n!(-\mathrm{res} - n + 1)} z^{-n}, $$

in which $z^\alpha$ for $\alpha \in \mathbb{C} \setminus \mathbb{Z}$ is to be understood as $\exp(\alpha \log z)$, where $\log z$ is well-defined if $z \in \tilde{\mathbb{C}}^*$. However, the absence of the log term does not imply anything on the isomorphism class of the double pole. With the choice of developing map $\phi$ above, the affine holonomy of the expression is $L: z \mapsto \lambda z + 0$ with $\lambda = \exp(-2\pi i \mathrm{res})$, in particular it fixes the origin. The question amounts to determining whether or not the limit of $\phi(x)$ as the real $x > 0$ tends to 0, is 0 or a non-zero complex number. With the expansion above, it seems hard.

Instead of using series expansions, we will try and determine the class using a path integral of $\phi' = z^{-\mathrm{res}} \exp(-1/z)$ as follows. Let $r > 0$ and consider the path $\gamma_r$ that is composed of three pieces: the first is the segment from 0 to $r$, the second is the circle of
center 0 and radius $r$ followed counterclockwise, and the last one is initial segment followed backwards. Then the double pole local isomorphism class is completely determined by whether or not the following integral vanishes:

$$I = \int_{\gamma_r} \phi'(z) dz$$

where $\phi'(z)$ is analytically continued along $\gamma_r$. By classical complex analysis, this integral is independent of $r$. Since as $x \to +\infty$, $\phi'(x) \sim x^{-\text{res}}$, we get that if $\text{Re}(\text{res}) > 1$, then the integral along the circle tends to 0 when its radius $r$ tends to infinity, hence $I$ is equal to the difference of the integral of two determinations of $\phi'$ along the real line:

$$\text{Re}(\text{res}) > 1 \implies I = (1 - e^{-2\pi i \text{res}}) \int_{0}^{+\infty} x^{-\text{res}} e^{-1/x} dx$$

and the change of variable $y = 1/x$ yields $I = (1 - e^{-2\pi i \text{res}}) \int_{0}^{+\infty} y^{\text{res}-2} e^{-y} dy$ and we recognize Euler's Gamma function. So $\text{Re}(\text{res}) > 1 \implies I = (1 - e^{-2\pi i \text{res}}) \Gamma_{\text{Euler}}(\text{res}-1)$. The quantity $\phi'(z)$ depends analytically on res and for a fixed $r$, the integral along $\gamma_r$ is analytic in res, hence $I$ is an analytic function of res. By the analytic identity theorem:

$$\forall \text{res} \in \mathbb{C} \setminus \mathbb{Z}, I = (1 - e^{-2\pi i \text{res}}) \Gamma_{\text{Euler}}(\text{res} - 1).$$

Thanks to the non trivial fact that the Gamma function never vanishes, it follows that

$$\forall \text{res} \in \mathbb{C} \setminus \mathbb{Z}, I \neq 0.$$

In these cases, the double pole is non-centered.

As a conclusion:

**Proposition.** For all $\text{res} \in \mathbb{C}$, for the unique (up to isomorphism) affine surface in the class $\mathcal{C}$ where the double pole $p$ has residue $\text{res}$, then $p$ is centered if and only if $\text{res} \in \mathbb{Z} \cap [2, +\infty)$.

Again, Theorem 7.3 contains this result and gives more information.

Last, we can reverse the line of arguments: the analysis of this section plus the geometric argument of Proposition 7.4 give a new proof that Euler's Gamma function does not vanish on $\mathbb{C} \setminus \mathbb{Z}$.<fnref n="28" />

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<footnote n="28">²⁸Classical proofs include one via Euler’s reflection formula and several via infinite products. One may also use the recursion formula $\Gamma(z + 1) = z\Gamma(z)$ together with an integral estimate when $\text{Re}(z)$ is much bigger than $\text{Im}(z)$. The reflection formula gives the alternate expression $I = -2i/(e^{\pi i \text{res}} \Gamma_{\text{Euler}}(2 - \text{res}))$.</footnote>
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(Xavier Buff) INSTITUT MATHÉMATIQUE DE TOULOUSE; UMR 5219; UNIVERSITÉ DE TOULOUSE; CNRS; UPS, F-31062 TOULOUSE CEDEX 9, FRANCE

Email address: xavier.buff@math.univ-toulouse.fr

(Arnaud Chéritat) INSTITUT MATHÉMATIQUE DE TOULOUSE; UMR 5219; UNIVERSITÉ DE TOULOUSE; CNRS; UPS, F-31062 TOULOUSE CEDEX 9, FRANCE

Email address: arnaud.cheritat@math.univ-toulouse.fr

(Guillaume Tahar) BEIJING INSTITUTE OF MATHEMATICAL SCIENCES AND APPLICATIONS, No. 544, HEFANGKOU VILLAGE, HUAIBEI TOWN, HUAIROU DISTRICT, BEIJING, CHINA, 101408

Email address: guillaume.tahar@bimsa.cn