\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 122, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/122\hfil Normal forms for holomorphic vector fields]
{Normal forms for singularities of one dimensional holomorphic vector fields}
\author[A. Garijo, A. Gasull, X. Jarque\hfil EJDE-2004/122\hfilneg]
{Antonio Garijo, Armengol Gasull, Xavier Jarque} % in alphabetical order
\address{Antonio Garijo \hfill\break
Dep. d'Eng. Inform\`atica i Matem\`atiques\\
Universitat Rovira i Virgili\\
Av. Pa\"{\i}sos Catalans, 26\\
43007 Tarragona, Spain}
\email{agarijo@etse.urv.es}
\address{Armengol Gasull \hfill\break
Dept. de Matem\`{a}tiques \\
Universitat Aut\`{o}noma de Barcelona \\
Edifici C, 08193 Bellaterra, Barcelona, Spain}
\email{gasull@mat.uab.es}
\address{Xavier Jarque \hfill\break
Dep. de Matem\`atica Aplicada i An\`alisi\\
Universitat de Barcelona\\
Gran Via 585\\
08007 Barcelona, Spain}
\email{xavier.jarque@ub.edu}
\date{}
\thanks{Submitted June 23, 2004. Published October 15, 2004.}
\thanks{Research supported by grants: BFM2002-04236-C02-2 from DGES,
BFM2002--01344 \hfill\break\indent
from DGES, 2001SGR-00173 from CONACIT, and
AYA2001-0762 from DGI}
\subjclass[2000]{34C20, 34A34, 32A10, 37C10}
\keywords{Meromorphic vector field; holomorphic vector field; normal form}
\begin{abstract}
We study the normal form of the ordinary differential equation
$\dot z=f(z)$, $z\in\mathbb{C}$, in a neighbourhood of a point
$p\in\mathbb{C}$, where $f$ is a one-dimensional holomorphic function
in a punctured neighbourhood of $p$. Our results include all cases except
when $p$ is an essential singularity. We treat all the other
situations, namely when $p$ is a regular point, a pole or a zero of
order $n$. Our approach is based on a formula that uses the flow
associated with the differential equation to search for the change of
variables that gives the normal form.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction and Main Result}
This note provides a proof for the following result:
\begin{theorem}\label{Main}
Let $f(z)$ be a one-dimensional holomorphic function in a punctured
neighbourhood $\mathcal{U}\subset\mathbb{C}$ of a point $p$. Consider the
ordinary differential equation
\begin{equation}\label{equacio}
\frac{dz}{dt} \, = \, f(z),\quad z\in \mathcal{U}, \quad
t\in\mathbb C.
\end{equation}
Then, in a neighbourhood of $p$, this equation is
conformally conjugate, in a neighbourhood of the origin, to
\begin{enumerate}
\item[(a)]\label{regular} $\dot z=1$, if $f(p)\ne0$,
\item[(b)]\label{cero} $\dot z=f'(p)z$, if $p$ is a zero of $f$ of order 1
({\it i.e.} $f'(p)\ne 0$),
\item[(c)]\label{degenerat} $\dot z=z^n/(1+cz^{n-1})$, where
$c=\mathop{\rm Res}(1/f,p)$, if $p$ is a zero of $f$ of order $n>1$,
\item[(d)]\label{pols} $\dot z=1/z^n$, if $p$ is a pole of order $n$.
\end{enumerate}
\end{theorem}
Statements (a), (b) and (c) are well-known, see for instance
\cite{BT}. To our knowledge statement (d), as it is presented
here, is new. Just topological conjugacy between equation
(\ref{equacio}) and $\dot z=1/z^n$ has been proved in
\cite{Ha1,Ha2, NK, Sv}. In \cite{GS} a similar result is also given. Our
approach is based in the one dimensional version of a nice
formula, as far as we know, firstly introduced in the unpublished
thesis of Pazzi \cite{P} and recently used in
\cite{{CMR,T1,T2}}. This formula allows to look for the conjugacy
between the original differential equation (\ref{equacio}) and its
normal form by using the flow of the primer. In Lemma~\ref{lema},
for the sake of completeness, we prove it in the one dimensional
framework. Notice that the case of $f$ having an essential
singularity at $p$ is not covered by the above theorem. In a
forthcoming paper we will consider this situation.
Also we observe that by using Theorem \ref{Main} it is easy to
describe the phase portrait, for $t\in\mathbb{R}$, of meromorphic
equations in a small punctured neighbourhood of their
singularities and poles. Furthermore, except for the simple zeroes
of $f$, it can be shown that the index at the point fully
characterizes the dynamics near the singularity.
\section{Proof of main Theorem}
Without loss of generality, we assume that the point $p$ in Theorem
\ref{Main} is moved to the origin. The solution of
(\ref{equacio}) passing through $z\in \mathcal U$ at $t=0$ is denoted by
$\varphi_f(t,z)$.
Let $f$ and $g$ be analytic functions in a punctured neighbourhood
of 0. It is said that the differential equations $\dot{z} = f(z)$
and $\dot{z}=g(z)$ are {\it conformally conjugated near the
origin} if there exists a conformal map $\Phi: \mathcal V \to
\mathcal W$, $\mathcal V$ and $\mathcal W$ being two open
neighbourhoods of 0, such that $\Phi(0)=0$ and
\begin{equation}\label{eq:ConfEqui}
\Phi(\varphi_f(t,z))=\varphi_g(t,\Phi(z)), \ \mbox{ for all }\ z\in
\mathcal V\setminus \{0\},
\end{equation}
and all $t$ for which the above expressions are well defined and
the corresponding points are in $\mathcal V$ and ${\mathcal W}$.
\begin{lemma}[\cite{CMR,T2}] \label{lema}
Let $z,w\in\mathbb C$ be given implicitly by the
equation $z=\varphi_f(h(w),w)$. Then if $h$ is a holomorphic
function, the expression of \eqref{equacio} in the
$w$-variable is
$$
\dot w=\frac{f(w)}{1+f(w)h'(w)} \, .
$$
\end{lemma}
\begin{proof}
By taking derivative with respect to $t$ of $z=\varphi_f(h(w),w)$, we obtain
$$
\dot z=\big(h'(w)\frac{\partial \varphi_f }{\partial t}(h(w),w)+
\frac{\partial \varphi_f}{\partial z}(h(w),w)\big)\dot w \ .
$$
Thus
\begin{equation}\label{eq2}
\dot z=\big(h'(w)f(z)+ \frac{\partial \varphi_f}{\partial
z}(h(w),w) \big)\dot w\ .
\end{equation}
To compute $\partial \varphi_f(t,z)/\partial z$ we note that it
is the solution of the Cauchy problem given by the variational
equation $\dot u=f'(\varphi_f(t,z))u$ with initial condition
$u(0)=1$. Observe that the function $f(\varphi_f(t,z))/f(z)$ is
the solution of this Cauchy problem. Hence by the uniqueness of
solutions we have
$$
\frac{\partial \varphi_f}{\partial z}(t,z)=\frac{f(\varphi_f(t,z))}{f(z)}.
$$
Evaluating the above expression at $(t,z)=(h(w),w)$ we obtain
$$
\frac{\partial \varphi_f}{\partial z}(h(w),w)=\frac{f(z)}{f(w)}.
$$
Consequently (\ref{eq2}) becomes
$$
f(z)=\big(h'(w)f(z)+ \frac{f(z)}{f(w)} \big)\dot w,
$$
as desired.
\end{proof}
This lemma points out that a suitable choice for the function $h$
may reduce equation (\ref{equacio}) to its normal form, as it is
stated in Theorem \ref{Main}. Before proving the theorem we show,
in two steps, a useful condition for two differential equations to
be conformally conjugated.
\begin{lemma}\label{lemma:tempswelldef}
The equation $z=\varphi_f(t,w)$, associated to the flow of
\eqref{equacio} is, in a punctured neighbourhood of the
origin, implicitly given by the equation
\begin{equation}\label{eq:tempsbendef}
\exp \Big(\int_w^z \frac{1}{cf(s)}\ ds \Big) =
\exp \big(\frac{t}{c}\big),\quad \mbox{when }
c:=\mathop{\rm Res}(1/f,0)\ne0,
\end{equation}
or by
\begin{equation}\label{eq:tempsbendefhol}
\int_w^z \frac{1}{f(s)}\ ds = t,\quad \mbox{when }
\mathop{\rm Res}(1/f,0)=0.
\end{equation}
Observe that the case $\mathop{\rm Res}(1/f,0)=0$ also includes the
situation in which $1/f$ is holomorphic at $z=0$.
\end{lemma}
\begin{proof}
To show the first statement, where $c\ne0$, we claim that, on one
hand, both sides of equation (\ref{eq:tempsbendef}) are well
defined in a punctured neighbourhood of the origin, and, on the
other hand, the solution, $z=\eta(t,w)$ (such that $\eta(0,w)=w$),
of the equation (\ref{eq:tempsbendef}) is indeed the flow
associated to equation (\ref{equacio}).
To see the claim we integrate $\dot z = f(z)$ in the ``formal" way to obtain
\begin{equation}\label{formal}
\int_w^z \frac{1}{f(s)}\ ds = t.
\end{equation}
The left-hand side of the above equation is not well defined since
its value depends, in general, of the path from $w$ to $z$ in
$\mathbb C$. In other words, the integral is only defined in a
neighbourhood of $z=0$ slit along the negative real line. However,
from the Residue Theorem, we know that such difference is either
zero, if the two paths do not cross the negative real line, or it
is $\pm 2\pi i$ times the residue of $1/f$ at $z=0$, if they
enclose $z=0$. Thus, dividing by $c\ne0$ and taking exponential,
we uniquely determine the value of the left hand side of equation
(\ref{eq:tempsbendef}) and, moreover, it has meaning in a whole
punctured neighbourhood of $z=0$ (including the negative real
line).
To complete the proof of the claim, we need to show that the solution,
$z=\eta(t,w)$ (such that $\eta(0,w)=w\ne0$), of the equation
(\ref{eq:tempsbendef}) is the flow associated to equation
(\ref{equacio}). To do so we use the Implicit Function Theorem. If
we denote by $F(z,w,t)=0$ the equation (\ref{eq:tempsbendef}) we
note that
$$
\frac{\partial}{\partial z}F(z,w,t)= \exp \Big(\int_w^z
\frac{1}{cf(s)}\ ds \Big) \frac{1}{cf(z)} \ne 0.
$$
Hence we know of the existence of $z=\eta(t,w)$ with $\eta(0,w)=w$. Now, if we
compute $\frac{\partial}{\partial t} \eta (t,w)$, we easily get
$$
\exp \Big(\int_w^{\eta(t,w)} \frac{1}{cf(s)} \, ds\Big)
\frac{1}{c f(\eta(t,w))} \frac{\partial}{\partial t} \eta (t,w) =
\frac{1}{c} \exp (\frac{t}{c}),
$$
or, equivalently,
$\frac{\partial}{\partial t} \eta (t,w) = f(\eta(t,w))$, as desired.
The latter statement of the lemma, {\it i.e.} when $\mathop{\rm Res}(1/f,0)=0$,
follows from the fact that, under this hypothesis, the left-hand
side of equation (\ref{formal}) is well defined in a punctured
neighbourhood of the origin.
\end{proof}
\begin{proposition}\label{teo2}
Let $f(z)$ and $g(z)$ be holomorphic functions in a punctured
neighbourhood of the origin and set $c={\rm Res}(1/f,0)$. Assume
that the function
\begin{equation}\label{condicio}
H(z):=\frac{f(z)-g(z)}{f(z)g(z)}=\frac 1{g(z)}-\frac1{f(z)}
\end{equation}
is a holomorphic map at $z=0$, and define $h(z)=\int_0^z
H(s)\,ds$. Then the transformation $z=\varphi_f(h(w),w)$, which
is not necessarily bijective, transforms the equation $\dot
z=f(z)$ into the equation $\dot w=g(w)$. Moreover, in a punctured
neighbourhood of the origin, the transformation can be implicitly
written as
\begin{equation}\label{canvifinal}
\exp \Big(\int_w^z\frac1{cg(s)}\,ds\Big)
= \exp \Big(\frac{1}{c} \int_0^z\big(\frac1{g(s)}
-\frac1{f(s)}\big)\,ds \Big),
\end{equation}
when $c\ne0$ or as
\begin{equation}\label{canvifinalc=0}
\int_w^z\frac1{g(s)}\,ds= \int_0^z\big(\frac1{g(s)}
-\frac1{f(s)}\big)\,ds ,
\end{equation}
when $c=0$. Moreover, this later equation simplifies to
\begin{equation}\label{canvifinal2}
\int_0^w\frac1{g(s)}\,ds= \int_0^z\frac1{f(s)}\,ds ,
\end{equation}
if $1/g(z)$ and $1/f(z)$ are both holomorphic at the origin.
\end{proposition}
\begin{remark} \rm
Note that by using the above proposition, but interchanging
the roles of $f$ and $g$, it can be shown that the local inverse
of the map $z=\varphi_f(h(w),w)$ is given by
$w=\varphi_g(-h(z),z)$.
\end{remark}
\begin{corollary}\label{coro}
Under the hypotheses of Proposition \ref{teo2},
let $w=W(z)$ be a solution, in a punctured neighbourhood of the
origin, of \eqref{canvifinal} if $c\ne0$ or of
\eqref{canvifinalc=0} if $c=0$, satisfying that $W(z)$ tends to 0
when $z$ tends to 0. If this solution can be extended to a
conformal map in a full neighbourhood of the origin, then the
ordinary differential equations
$$
\frac{dz}{dt} = f(z),\quad \mbox{and}\quad \frac{dz}{dt} = g(z),
$$
are conformally conjugated near the origin.
\end{corollary}
\begin{proof}[Proof of Proposition \ref{teo2}]
We define $w=W(z)$ as a solution
of $z=\varphi_f(h(w),w)$, satisfying $W(0)=0$, where $h$ is an
unknown function. By Lemma~\ref{lema}, the suitable condition over
$h$ which implies that the expression of $\dot z=f(z)$ in the
variable $w$ is $\dot w=g(w)$, reads as
$$
h'(w)=H(w)=\frac{f(w)-g(w)}{f(w)g(w)}=\frac1{g(w)}-\frac1{f(w)}.
$$
By hypothesis, the right hand side of the above equation is
holomorphic. Hence choosing $h(w)=\int_0^w H(s)\,ds$, the
transformation given by $z=\varphi_f(\int_0^w H(s)\,ds,w)$,
transforms $\dot z=f(z)$ into $\dot w=g(w)$. When
Res$(1/f,0)=c\ne0$, we apply equation (\ref{eq:tempsbendef}) in
Lemma \ref{lemma:tempswelldef} to write equation
$z=\varphi_f(t,w)$ as
$$
\exp \Big(\int_w^z \frac{1}{cf(s)}\ ds \Big) = \exp (\frac{t}{c}).
$$
Hence equation $z=\varphi_f(h(w),w)$ writes as
$$
\exp \Big(\int_w^z \frac{1}{cf(s)}\ ds \Big)
= \exp \big(\frac{h(w)}{c}\big)
= \exp \Big(\frac{1}{c}\int_0^w \big(\frac1{g(s)}-\frac1{f(s)}\big)\ ds \Big).
$$
which is equivalent to (\ref{canvifinal}) by using that
$$
\int_0^w \big(\frac1{g(s)}-\frac1{f(s)}\big)\,ds
= \int_0^z \big(\frac1{g(s)}-\frac1{f(s)}\big)\,ds
+\int_z^w \big(\frac1{g(s)}-\frac1{f(s)}\big)\,ds .
$$
When $c=0$ we can perform similar computations but using
(\ref{eq:tempsbendefhol}) instead of (\ref{eq:tempsbendef}) to
arrive to the desired result. Finally if both, $1/f(z)$ and
$1/g(z)$ are holomorphic at zero, easy manipulations transform
equation (\ref{canvifinalc=0}) into (\ref{canvifinal2}).
\end{proof}
It only remains to prove the main theorem.
\begin{proof}[Proof of Theorem~\ref{Main}]
We deal first with statements (a) and (d),
{\it i.e.} for $f$ having either a regular point ($n=0$) or a pole
of order $n$
at the origin. Hence $n\in\mathbb{N}\cup\{0\}$. In these cases
$f(z)=1/(z^nG(z))$ with
$G(0)=g_0\ne0$. We choose $g(z)=1/(g_0z^n)$. Notice that
$H(z)=z^n(g_0-G(z))$ is a holomorphic map at $z=0$ and we are
under the hypotheses of Proposition~\ref{teo2}. Furthermore, since
$1/f(z)$ and $1/g(z)$ are also holomorphic, again by
Proposition~\ref{teo2}, the transformation $z=\varphi_f(h(w),w)$,
writes as (\ref{canvifinal2}), which is
$$
\int_0^w g_0s^n \,ds=\int_0^z s^nG(s)\,ds.
$$
Thus
$$
w=W(z):=z \sqrt[n+1]{\frac{(n+1)\int_0^z s^nG(s)\,ds}{g_0z^{n+1}}},
$$
is a conformal change of variables (notice that $W(0)=0$ and
$W'(0)\ne 0$) between $\dot z=1/(z^nG(z))$ and
$\dot w=1/(g_0w^n)$. A new change of variables $w\to\alpha w$,
for a convenient $\alpha\in\mathbb{C}$, finishes the proof of
cases (a) and (d).
We consider now statement (b). Write $f(z)=z/G(z)$, with
$G(0)=g_0\ne0$ and $g(z)=z/g_0$. Thus $H(z)$ is a holomorphic map
at $z=0$ and we are again under the hypotheses of
Proposition~\ref{teo2}. Arguing like in the precedent case, but
using equation (\ref{canvifinal}) instead of (\ref{canvifinal2}),
we get that $z=\varphi_f(h(w),w)$ writes as
$$
\exp \Big(\int_w^z \frac{1}{s}\,ds\Big)
= \exp \Big(\frac{1}{g_0}\int_0^z\frac{g_0-G(s)}s\,ds \Big).
$$
It is easy to see that the function
$$
w=W(z):=z\exp \Big( \int_0^z\frac{G(s)-g_0}{g_0s}\,ds\Big),
$$
is a solution of the above equation. Furthermore it is a conformal
map and satisfies $W(0)=0, W'(0)\ne0$. Hence by Corollary
\ref{coro} it is a conformal conjugacy between $\dot z=z/G(z)$
and $\dot w=w/g_0$, as stated.
Finally we deal with statement (c). In this case $f(z)=z^n/G(z)$
where $n\ge2$ is an integer number, and $G(0)=g_0\ne0$. As
usually, if $G(z)=g_0+g_m z^m+O(m+1)$, by considering a conformal
change of variables of the form $z=z_1(1+az_1^m)$ we get $\dot
z_1=z_1^n/(g_0+[(m+1-n)ag_0+g_m]z_1^m+O(m+1))$. Hence,
if $1\le m\le n-2$, we can choose the parameter $a$ in such a way
that $\dot z_1=z_1^n/(g_0+O(m+1))$, and so, it is not restrictive
to assume that $f(z)=z^n/(g_0+g_{n-1}z^{n-1}+\tilde G(z))$,
$\tilde G(z)/z^n$ being analytic at the origin. We consider
$g(z)=z^n/(g_0+g_{n-1}z^{n-1})$, and we proceed as in the previous
cases. It is convenient, however, to deal with the cases ${\rm
Res}(1/f,0)=g_{n-1}\ne0$ and $g_{n-1}=0$ separately. We start with
the case $g_{n-1}\ne0$.
By using (\ref{canvifinal}) we get that $z=\varphi_f(h(w),w)$
writes as
\begin{equation}\label{slit0}
\exp \Big(\int_w^z \frac{g_0+g_{n-1}s^{n-1}}{g_{n-1}s^n}\,ds\Big)
= \exp \Big(\int_0^z\frac{-\tilde G(s)}{g_{n-1}s^n}\,ds\Big).
\end{equation}
Although the above equation is well defined in a punctured
neighbourhood $\mathcal N$ of the origin, in order to take
logarithms (principal determination), we consider it restricted to
$\mathcal{N}$ slit along the negative real line. Thus, it writes
as
\begin{equation}\label{slit}
\frac{g_0}{1-n}\big[\frac{1}{z^{n-1}}-\frac1{w^{n-1}}
\big]+g_{n-1} \log\frac zw+\int_0^z\frac{\tilde
G(s)}{s^n}\,ds=0,
\end{equation}
We note that if $g_{n-1}=0$, using (\ref{canvifinalc=0}) instead
of (\ref{canvifinal}), we would have arrived also to (\ref{slit}).
So, in what follows, we proceed with the two cases together. By
multiplying the above equation by $z^{n-1}$ and introducing the
new variable $u=z/w$ we get the equation
$$
F(u,z)=\frac{g_0}{1-n}\left[1-u^{n-1} \right]+g_{n-1}z^{n-1} \log
u+z^{n-1}\int_0^z\frac{\tilde G(s)}{s^n}\,ds=0.
$$
Note that $F(1,0)=0$ and $\partial F (1,0)/\partial u\ne0$.
Therefore, by the Implicit Function Theorem, near $(u,z)=(1,0)$,
there exists a unique analytic function $u=U(z)$ such that
$F(U(z),z)\equiv0$ and $U(0)=1$. Hence $w=W(z):=z/U(z)$ is a
conformal solution of (\ref{slit}) on $\mathcal{N}$ slit along the
negative real line, which is indeed a solution of (\ref{slit0}) on
$\mathcal{N}$. Furthermore, since it is also conformal at the
origin and $W(0)=0$, by using Corollary \ref{coro} we know that it
gives a conformal change of variables between the equations $\dot
z=z^n/(g_0+g_{n-1}z^{n-1}+\tilde G(z)) $ and $\dot
w=w^n/(g_0+g_{n-1}w^{n-1}). $
By introducing a final change of the form $w\to\alpha w$
we can impose that $g_0=1$ and the expression of the normal form
given in the theorem follows. It is easy to check that ${\rm
Res}(1/f,0)=g_{n-1}$ is an invariant for conformal conjugation,
see Remark~\ref{nota1} or \cite{BT}. Hence $c$ is ${\rm
Res}(1/f,0)$ and the proof of statement (c) in Theorem \ref{Main}
follows.
\end{proof}
\begin{remark} \label{nota1} \rm
As we have already mentioned, in \cite{BT} it is proved
that under the hypotheses of Proposition~\ref{teo2} a necessary
condition for equations $\dot{z}=f(z)$ and $\dot{z}=g(z)$ to be
conformally conjugated near the origin is that
$\mathop{\rm Res}(1/f,0)=\mathop{\rm Res}(1/g,0)$. Notice that the condition
\eqref{condicio} in Proposition~\ref{teo2}, implies, among other
things, that ${\rm Res}(1/f,0)={\rm Res}(1/g,0)$.
\end{remark}
\begin{remark} \rm
Observe that in case that both $\dot z=f(z)$ and $\dot z=g(z)$
have a pole at the origin (not necessarily of the same order), the
map given by (\ref{canvifinal2}) transforms one equation into the
other. However, this transformation is not bijective (so, it is
not a change of variables) unless both poles have the same order.
\end{remark}
\begin{remark} \rm
In case (c) of Theorem~\ref{Main}, it is not difficult to find other
simple models for the normal form. For instance, the one given in
the statement can be replaced by $\dot{z}=z^n - c^{1/(n-1)}
z^{n+1}$ or by $\dot z= z^n-cz^{2n-1}$.
\end{remark}
\begin{remark} \rm
The main results given in this paper remain true for real analytic
vector fields. Only some constants have to be added in the real
normal forms.
\end{remark}
\subsection*{Concluding Remarks}
If we are not interested in a constructive proof of items (a), (b)
or (d) of Theorem~\ref{Main} we can check the following
assertions, which give a straightforward proof (modulus a final
rescalling of the variable $w$ in cases (a) and (d)):
\begin{enumerate}
\item[(i)] For $n\in\mathbb{N}\cup\{0\}$ the conformal change of variables
$$
w=z\sqrt[n+1]{\frac{(n+1)\int_0^z s^nG(s)\,ds}{g_0z^{n+1}}},
$$
transforms $\dot z=1/(z^nG(z))$ with $G(0)=g_0\ne0$ into $\dot w=1/(g_0w^n)$.
\item[(ii)] The conformal change of variables
$$
w=z\exp\Big( \int_0^z\frac{G(s)-g_0}{g_0s}\,ds\Big),
$$
transforms $\dot z=z/G(z)$ with $G(0)=g_0\ne0$ into $\dot w=w/g_0$.
\end{enumerate}
As it has been shown in the previous section, the change of
variables which proves item (c) is not explicit.
\subsection*{Acknowledgements}
The authors want to thank Jordi Villadelprat for letting
us know about the formula given in Lemma~\ref{lema}.
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