\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 55, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/55\hfil Painlev\'e's determinateness theorem]
{Painlev\'e's determinateness theorem extended to proper coverings}
\author[C. Meneghin\hfil EJDE-2010/55\hfilneg]
{Claudi Meneghin}
\address{Claudi Meneghin \newline
(Isrc) - Fermo Posta Chiasso 1, CH-6830 Chiasso, Switzerland}
\email{claudi.meneghin@gmail.com}
\thanks{Submitted January 23, 2009. Published April 19, 2010.}
\subjclass[2000]{34M35, 34M45}
\keywords{Painlev\'e's determinateness theorem;
complex differential equations; \hfill\break\indent
Cauchy's problem; analytic continuation;
singularities; multi-valued function; Riemann surface}
\begin{abstract}
We extend Painlev\'e's determinateness theorem to first-order
ordinary differential equations in the complex domain,
with known terms allowed be multivalued in the dependent variable.
The multivaluedness is supposed to be resolved by proper coverings.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction}
What is generally referred to as
\emph{Painlev\'e's Determinateness Theorem}
\cite[Thm. 3.3.2]{hille} for first order ordinary differential
equations in the complex domain is stated as follows:
\begin{quote}
If $F(z,w)$ is a rational function of $w$ with coefficients which
are algebraic functions of $z$, then any movable singularities of
the solutions to the first order ODE $w' = F(z,w)$ are poles and/or
algebraic branch points.
\end{quote}
The so called ``known term'' $F$ is required to be single-valued
in the dependent variable $w$; the goal of this note is to allow $F$
to be multivalued in $w$, and conclude that none of the movable
singularities are essential notwithstanding.
The multivaluedness of $F$ will be resolved by passing to a
Riemann domain $(\Delta,p)$ over $\mathbb{C}^2$ minus a
complex-analytic curve,
with the main assumption that $(\Delta, p)$ be
a \emph{proper cover}
of $p(\Delta)$.
This hypothesis cannot be dropped in general, as example
\eqref{noproper} shows.
More singularities (not necessarily poles) for $F$ will be allowed
on a complex-analytic curve in $\Delta$.
Previous statements of Painlev\'e's theorem \cite[p. 38]{painleve'},
\cite[p. 327-328]{picard}, \cite[p. 292]{ince}, \cite[Thm. 3.3.1]{hille},
read as follows:
\begin{quote}
If a solution of the first-order ODE $w' = F(z,w)$ is continued
analytically along a rectifiable arc from $z=z_0$ to $z=z_1$
avoiding the set $S$ of fixed singularities, and if $z_1\not\in S$,
then the solution tends to a definite limit, finite or infinite,
as $z\to z_1$.
\end{quote}
The two statements of Painlev\'e's determinatess theorem are equivalent
under the assumption about the known term $F$
(see the references here above);
since we make a broader hypothesis, we can no longer take equivalence
for granted; in particular, logarithmic branch points in the solutions
cannot be excluded, as pointed out in \cite[sec. 3.1]{hille}
(see also example \ref{Solution with logarithmic singularity} below).
In this note we generalize this latter version, but the final remark
shows that movable essential singularities can be ruled out anyway;
the question whether or not, in our broader setting, natural boundaries
could arise will be the object of future investigations.
This note does not require Bieberbach's precise
definition of a fixed or of a movable singularity
(\cite{bieberbach}, quoted in \cite{steinmetz});
we will use these notions in an informal fashion
(as in \cite{hille,ince, painleve',picard}), within the examples only.
Before stating and proving our main theorem,
we now introduce some terminology and discuss some examples.
\section{Terminology}
A \emph{Riemann domain} over a region
$\mathcal{U}\subset\mathbb{C}^n$ is a complex manifold $\Delta$ with
an everywhere maximum-rank holomorphic surjective mapping
$p:\Delta\to\mathcal{U}$; $\Delta$ is \emph{proper} provided that so is $p$
(see \cite[p. 43]{gunros});
a curve $\mathcal{S}\subset\mathbb{C}^n$ is \emph{complex-analytic}
provided that it is the common zero set of $N-1$ complex-analytic
functions on $\mathbb{C}^n$;
when $n=1$ we talk about \emph{Riemann surfaces}.
\begin{definition} \label{continuation} \rm
Let $M$ be a complex manifold, $U$ an open set in $\mathbb{C}^n$,
$f:U\to M$ an holomorphic mapping: a
{\rm regular analytic continuation} of the holomorphic mapping
element $(U,f)$, is a quadruple $(S,\pi,j,F)$ such that:
\begin{itemize}
\item[(1)] $S$ is a connected Riemann domain over a region
in $\mathbb{C}^n$;
\item[(2)] $\pi: S\to \mathbb{C}^n$ is an
everywhere nondegenerate holomorphic mapping such
that $U\subset \pi(S)$;
\item[(3)]
$j: U\to S$ is a holomorphic immersion such
that $\pi\circ j=id|_{U}$;
\item[(4)]
$F: S\to M$ is a holomorphic mapping such
that $F\circ j=f$.
\end{itemize}
Let $\gamma:I\to\mathbb{C}^n$ be an arc
(with $I=[0,1]$ or $I=[0,1)$)
such that $\gamma (0)=X$; a regular analytic continuation along
$\gamma$ of $(U,f)$ is a regular analytic continuation $(S,\pi,j,F)$
of $(U,f)$ such that there exists an arc
$\widetilde \gamma:I\to S $ with
$\pi \circ \widetilde \gamma=\gamma $.
\end{definition}
\subsection{Cauchy's problems with multivalued known terms}
\label{multivalued known terms}
Let us now focus on differential equations whose
``known terms'' are defined on Riemann domains rather than just
on open sets in $\mathbb{C}^2$. Introduce the following concepts:
\begin{itemize}
\item a complex-analytic curve
$\mathcal{B}=\{(z,w)\in\mathbb{C}^2: B(z,w)=0\}$,
with $ B$ holomorphic on $\mathbb{C}^2$, called the {\bf branch
locus} of the differential equation, and a proper Riemann domain
$(\Delta , p)$ over $\mathbb{C}^2\setminus\mathcal{B}$. Note that
we do not require $\mathcal{B}$ to be algebraic.
\item
a complex one-dimensional submanifold $\Sigma \subset \Delta$,
such that $p(\Sigma)$ is included
in a complex-analytic curve $\Lambda (z,w)=0$ in $\mathbb{C}^2$;
$p(\Sigma )$
will be referred to
as the {\bf singularity locus};
\item
the branch and the singularity loci will be collectively
referred to as {\bf the singularities of the differential
equation};
\item
a holomorphic function $F$ on $\Delta \setminus \Sigma$,
called the {\bf known term};
\item
a point $X_0\in \Delta \setminus{\Sigma }$,
with $(z_0,w_0):=p(X_0)$; we will refer to $w_0$ as the {\bf initial value}
of the Cauchy problem and to $z_0$ as the {\bf initial point}; we will also
refer to $(z_0,w_0)$ collectively as the {\bf initial values};
\item
a local inverse $\eta $ of $p$, defined in a
bidisc $\mathbb{D}_1\times\mathbb{D}_2 $ around $(z_0,w_0)$.
\end{itemize}
The above definitions are meant to be referred to a differential equation
(or to an associated Cauchy problem) and not to its solutions.
\section{Examples}
In the realm of practice, the usual symbols of multivalued
functions such as $\log$ or $\sqrt{\ }$ will
go on to be used as well as the attributes \emph{multi-valued} or
\emph{single-valued}.
This is perfectly rigorous (even by a geometric point of view),
inasmuch as the underlying
machinery of analytic continuation is understood; in particular,
a branch of the multivalued known term
will always have to be specified alongside the initial conditions;
i.e., a local inverse $\eta$ of the covering map
$p:\Delta\to\mathbb{C}^2$
will have to be explicitly chosen there.
\subsection{Attaining singularities of the known term}
\label{leading}
Consider the Cauchy problem
\begin{gather*}
w'(z)={\sqrt{(1-w^2(z))}\ \frac{w(z)}{\sin z}} \\
w(\pi/4 )=\sqrt{2}/2.
\end{gather*}
Here we understand the choice of the positive branch of the square root
corresponding to the initial values
$(\pi/4, \sqrt{2}/2)$.
In the terminology of section \ref{multivalued known terms}, we have:
\begin{itemize}
\item
the branch locus is $\mathcal{B}=\{(z,w)\in\mathbb{C}^2: {w=\pm 1\}}$,
the Riemann domain of the known term is
$\Delta=\{(z,w,y)\in\mathbb{C}^3:w^2+y^2=1,\ y\not=0\} $,
with the projection mapping
$p(z,w,y)=(z,w)$, a twofold covering, hence a proper mapping;
\item
the singularity locus is
$\Sigma = \{(z,w,y)\in\Delta:z=k\pi, k\in\mathbb{Z}\}$;
\item
the known term $F:\Delta \setminus \Sigma\to\mathbb{C}$ is defined by
$F(z,w,y)=yw/\sin(z)$;
\item
the lifted initial point is
$X_0=(\pi/4, \sqrt{2}/2, \sqrt{2}/2)\in\Delta\setminus\Sigma$;
note that $p(X_0)=(z_0,w_0)=(\pi/4, \sqrt{2}/2)$;
\item
$\eta (z,w)= (z,w,\sqrt{1-w^2})$, where the positive branch of the
square root has been chosen.
\end{itemize}
The singularities of the equation in the underlying $\mathbb{C}^2$ lie on
$\{z=k\pi\}\cup\{w=\pm 1\}$.
The problem is solved by the entire function
$w(z)=\sin(z)$ (clearly admitting analytic continuation and, a fortiori,
limit, everywhere in $\mathbb{C}$).
Note that the multivaluedness in $w$ of the known term of this Cauchy
problem makes it attain singularities along the graph of the solution,
more precisely at $z=\frac{\pi }{2}+k\pi$.
This fact does not affect the analytic continuation of the solution since
the above singularities can be avoided by continuing along a suitable
real arc; compare
the argumentation following \eqref{discrete}.
\subsection{A problem with essential singularities}
The Cauchy problem
\begin{gather*}
w'(z)=({e^{z\cdot w(z)}+1})^{-1}
({e^{-z\cdot w(z)}}-w(z))
e^{({e^{z\cdot w(z)}-z})^{-1}+1} \\
w(2)=0
\end{gather*}
is solved by
$w(z)=[\log(z-1)]/z$.
The known term of this problem is single valued on
$\mathbb{C}^2$, has poles on the complex-analytic
curve ${e^{wz}=-1}$ and essential singularities on
${e^{wz}=z}$. No line $z={\rm const}$
(in particular $z=1$) is a singularity.
In view of theorem \ref{painleve'}, note that $w$ can be analytically
continued
along the arc $\gamma $ defined on $[0,1)$ by $\gamma(t)=2-t$ and
there does exist $\lim_{t\to 1}[\log(1-t)]/(2-t)=\infty$.
\subsection{No limit}
\label{leading1}
Consider the Cauchy problem:
\begin{gather*}
w'(z)=-{\sqrt{1-w^2(z)}}/{z^2} \\
w((1+i)^{-1})=\sin(1+i+c).
\end{gather*}
Here we suppose $| c|$ small enough and understand the choice
of the positive branch of the square root
corresponding to the initial values $((1+i)^{-1}, \sin(1+i+c))$.
In the terminology of section \ref{multivalued known terms}, we have:
\begin{itemize}
\item
the branch locus is $\mathcal{B}=\{(z,w)\in\mathbb{C}^2: {w=\pm 1\}}$,
the Riemann domain of the known term is
$\Delta=\{(z,w,y)\in\mathbb{C}^3:w^2+y^2=1,\ y\not=0\} $, with the projection mapping
$p(z,w,y)=(z,w)$, a twofold covering, hence a proper mapping;
\item
the singularity locus is $\Sigma = \{(z,w,y)\in\Delta:z=0\}$;
\item
the known term $F:\Delta \setminus \Sigma\to\mathbb{C}$ is defined by
$F(z,w,y)=-y/z^2$;
\item
the lifted initial point is $X_0=(1/\pi, -\sin(c), -\cos(c))\in\Delta\setminus\Sigma$;
note that $p(X_0)=(z_0,w_0)=(1/\pi, -\sin(c))$;
\item
$\eta (z,w)= (z,w,-\sqrt{1-w^2})$, where the positive branch of the square root has been
chosen.
\end{itemize}
The singularities of the equation in the underlying $\mathbb{C}^2$ lie on
$\{z=0\}\cup\{w=\pm 1\}$.
The problem is solved by $w(z)=\sin(1/z+c)$, showing an essential
singularity at $z=0$.
Note that $w$ can be analytically continued
along the arc $\gamma $ defined on $[0,1)$ by
$\gamma(t)= (1-t)(1+i)^{-1}$ and there does not exist
$\lim_{t\to 1}\sin(1/\gamma(t)+c)$; in view of theorem \ref{painleve'},
this should be compared with the fact that $\{z=0\}\subset\mathbb{C}^2$
is a line of poles for the known term.
\subsection{Solution with logarithmic singularity}
\label{Solution with logarithmic singularity}
In the Cauchy problem
\begin{gather*}
w'(z)=e^{-w(z)}({1+\sqrt[3]{e^{w(z)}-z+1}})/2\\
w(1)=0,
\end{gather*}
we understand the choice of the positive branch of the cube root
corresponding to the initial values $(1,0)$.
In the terminology of section \ref{multivalued known terms}, we have:
\begin{itemize}
\item
the branch locus is $\mathcal{B}=\{(z,w)\in\mathbb{C}^2: e^w -z+1=0\}$,
the Riemann domain of the known term is
$\Delta=\{(z,w,y)\in\mathbb{C}^3:e^w-z+1=y^3, \ y\not=0\} $, with the projection mapping
$p(z,w,y)=(z,w)$, a threefold covering, hence a proper mapping;
\item
the singularity locus $\Sigma $ is empty, indeed
the known term $F:\Delta\to\mathbb{C}$, defined by
$F(z,w,y)=e^{-w}(1+y)/2$ is holomorphic on the whole of $\Delta$;
\item
the lifted initial point is $X_0=(1,0,1)\in\Delta$;
note that $p(X_0)=(z_0,w_0)=(1,0)$;
\item
$\eta (z,w)= (z,w,\sqrt[3]{e^{w(z)}-z+1})$, where the positive branch of the cube root has been
chosen.
\end{itemize}
The singularities of the equation in the underlying $\mathbb{C}^2$
lie on the curve $e^{w}-z=-1$.
The problem is solved by
$w(z)=\log z$, which can be analytically continued
along the arc $\gamma $ defined on $[0,1)$ by
$\gamma(t)=1-t$. In view of theorem \ref{painleve'}, note that the
complex line ${z=0}$
is not a singularity for the differential equation and
there does exist $\lim_{t\to 1}\log(1-t)=\infty$.
\subsection{A separable multivalued problem}
Consider the Cauchy problem
\begin{gather*}
w'(z)={\sqrt{z}}/{\sqrt{w(z)}} \\
w(1)=(1+c)^{2/3}.
\end{gather*}
We have supposed $| c|$ is positive, real and small enough; we have
chosen the positive branches of the square and cube roots
corresponding to the initial values
$(1,(1+c)^{2/3})$.
As in the preceding examples, the Riemann domain
of the known term ${\sqrt{z}}/{\sqrt{w}}$ is proper;
the underlying singularities of the equation are on
$\{z=0\}\cup\{w=0\}$.
The problem is solved by $w(z)=(z^{3/2}+c)^{2/3}$,
showing a fixed algebraic branch point at $z=0$ and a movable one
at $z^{3/2}+c=0$.
Note that $w$ can be analytically continued
along the arc $\gamma $ defined on $[0,1)$ by
$\gamma(t)=1-t$ (without stumbling on
any of the movable branch
points $z^{3/2}+c=0$ on this path,
since $(1-t)^{3/2}>0$ and $c>0$) and
$\lim_{t\to 1}[(1-t)^{3/2}+c)]^{2/3}= c^{2/3}$.
In a different fashion,
$w$ admits the following analytic continuation along a path
pointing towards one of the movable branch points:
consider the arc, defined on $[0,1]$:
$$
\beta(t):=
\begin{cases}
e^{4\pi it} & \text{if $0\leq t\leq 1/2 $}\\
2(1-t)+c^{2/3}(2t-1) & \text{if $1/2\leq t\leq 1 $}.
\end{cases}
$$
After that $t$ has run on $[0,1/2]$, the analytic continuation of
$w(z)=(z^{3/2}+c)^{2/3}$ has changed sign from positive to negative;
further running the parameter on $[1/2,1]$ makes $z$ run into $c^{2/3}$,
which is this time a branch point for
$(z^{3/2}+c)^{2/3}$ on this path,
since $[2(1-t)+c^{2/3}(2t-1)]^{3/2}<0$ and $c>0$.
All the same, notice that
$\lim_{t\to 1}\{[2(1-t)+c^{2/3}(2t-1)]^{3/2}+c\}^{2/3}=0$
on this branch i.e., the analytic continuation of the solution of
our Cauchy problem admits limit even in the above circumstance.
\subsection{Counterexample: A logarithmic known term}
Consider the autonomous Cauchy problem
\begin{equation}
\begin{gathered}
w'(z)=-w(z) \log^2(w(z)) \\
w(0)=e^{-1/c}.
\end{gathered}
\label{noproper}
\end{equation}
This problem is solved by $w(z)=e^{1/(z-c)}$. Notice that
the multivaluedness in $w$ of the known term is logarithmic.
In the terminology of section \ref{multivalued known terms}, we have:
\begin{itemize}
\item
the branch locus is $\mathcal{B}=\{(z,w)\in\mathbb{C}^2: w=0\}$,
the Riemann domain of the known term is
$\Delta=\{(z,w,y)\in\mathbb{C}^3:w=e^y\} $, with the projection mapping
$p(z,w,y)=(z,w)$, which is not a proper covering;
\item
the singularity locus $\Sigma $ is empty, indeed
the known term $F:\Delta\to\mathbb{C}$, defined by
$F(z,w,y)={-w}y^2$ is holomorphic on the whole of $\Delta$;
\item
the lifted initial point is $X_0=(0, e^{-1/c}, -1/c)\in\Delta$;
note that $p(X_0)=(z_0,w_0)=(0, e^{-1/c})$;
\item
$\eta (z,w)= (z,w,\log w)$, where the real branch of the logarithm has been
chosen.
\end{itemize}
The singularities of the equation in the underlying $\mathbb{C}^2$
lie on the curve ${w}=0$.
In view of theorem \ref{painleve'}, note that the known term is not
resolved by a proper cover and that the solution has a movable essential
singularity; alternatively,
it can be stated that there exists a path $\gamma$ defined on $[0,1]$,
joining $0$ and $c$ and such that
$w$ admits analytic continuation along
$\gamma|_{[0,1)}$ but no continuous extension up to $\gamma(1)=c$.
\medskip
\noindent
{\bf Remark} This example shows that, in general, the hyphotesis
in theorem \ref{painleve'} that the
multivaluedness of the known term be resolved by a proper cover cannot
be dropped; however,
a first order ODE with nonproper covering associated to its known term
can yield a family of functions free from movable singularities
notwithstanding.
For instance, the equation
$$
w'(z)=\sqrt{1-w^2(z)}\ \frac{\arcsin(w(z))}{z}
$$
admits the family of solutions $\{\sin(kz)\}_{k\in\mathbb{C}}$,
which are free from any singularities at all.
\section{Main result}
Now we are ready to introduce the main issue of this paper:
in the terminology of section \ref{multivalued known terms},
introduce the (well defined) Cauchy problem:
\begin{equation}
\begin{gathered}
u'(v) =F\circ\eta(z, w(z)) \\
w(z_0)=w_0.
\end{gathered} \label{cauchy}
\end{equation}
Note that this problem is of a classical type; i.e.,
the known term is defined on an open set $\mathcal{U}\subset\mathbb{C}^2$
and a solution is sought that be a holomorphic function on a
one-dimensional complex disc and whose graph is contained in $\mathcal{U}$.
Thanks to the classical existence-and-uniqueness theorem
(see \cite[Thm. 2.2.2]{hille}, \cite[p.281-284]{ince}, such a solution
does exist.
The problem of its analytic continuation is natural and settles
(besides the usual matters dealing with
the analytic continuation of a function of one complex variable)
a supplementary
question; i.e., what happens if the analytic continuation $\omega$
of the graph of the solution leads to
singularities in the known term; i.e., for instance, points where
$F\circ\eta $ is not holomorphic?
Let $\gamma$ be an arc defined on $[0,1]$: if the Riemann domain
$(\Delta, p)$ resolving the multivaluedness of $F$ is proper
and the complex line $\{w=\gamma(1)\}$ is not a singularity,
Theorem \ref{painleve'} answers that
the feasibility of analytic continuation along $\gamma$ restricted
to the semi-open interval $[0,1)$
entails the existence of a (finite or infinite) limit for
$\omega\circ\gamma$ as the arc parameter tends to $1$.
Now we need a technical lemma.
\begin{lemma} \label{fmetric}
Let $X$ be a metric space, $\alpha : [0,1)\to X$ a continuous
arc such that $\lim_{t\to 1}\alpha (t)$ does not exist
in $X$.
\begin{itemize}
\item[(A)] let $\{x_l\}\to x_{\infty }$ be an injective converging
sequence in $X$: then there exists a
sequence $\{t_k\}\to 1$ and an open neighbourhood $U$
of $\{x_l\}\cup \{x_{\infty }\}$ such that
$\{\alpha (t_k)\}\subset X\setminus U$.
\item[(B)] for every $N$-tuple $\{x_1\dots x_N\}\subset X$ there
exists a sequence $\{t_i\}\to b$ and neighbourhoods
$U_k$ of $x_k$ such that $\{\alpha (t_i)\}\subset X\setminus
\bigcup_{k=1}^N U_k$.
\end{itemize}
\end{lemma}
\begin{proof}
(A) Since none of the $\{x_l\}$'s ($l\in\mathbb{N}\cup \{\infty \}$)
is $\lim_{t\to 1}\gamma(t) $, we have that for every
$l\in\mathbb{N}\cup \{\infty \}$
there exists an open neigbhourhooud
$V_l$ of $x_l$ such that
$\alpha ([\lambda ,1))\not\subset V_l$
for every $\lambda \in[0,1)$; moreover, up to shrinking $V_{\infty }$,
there exists $N>1$ such that $n>N$ implies $x_n\in V_{\infty }$
but $x_N\not\in V_{\infty }$. Clearly we can choose the
$\{V_l\}$'s in such a way that: $V_i\cap V_k=\emptyset$
if $i,k\in\mathbb{N}$ and $i\not=k$; $V_i\cap V_{\infty }=\emptyset $
if $l\leq N$.
Let now $U:=(\bigcup_{l=1}^N V_l)\cup V_{\infty }$:
by construction, $U$ is disconnected.
Since $\alpha ([\lambda ,1 )) $ is, by contrast, connected, and,
by construction, $\alpha ([\lambda ,1 )) $
is not contained in a single connected component of $U$,
we must have $\alpha ([\lambda ,1))\not\subset U$; this entails that,
for every $k>0$, the set
$W_k:=\alpha^{-1}(X\setminus U)\cap(1-1/k,1)$ is not empty;
picking $t_k \in W_k$ ends the proof.
The proof of (B) is analogous and will be omitted.
\end{proof}
We also need to generalize Hille's theorem \cite[Thm 3.2.1]{hille}
to our broader setting.
We will use once more the notation of section
\ref{multivalued known terms}.
Note that in the proof of this lemma we are forced to work first in
the underlying environments
$\mathbb{C}$ and $\mathbb{C}^2$ and to lift the results by local
charts into the overlying Riemann surface $R$ and domain $\Delta$.
This is why derivation is defined on $\mathbb{C}_z$ and the analytic
continuation of the solution
takes values in $\mathbb{C}_w$. Also, the Taylor developments
\eqref{taylor} are feasible using local charts in $\mathbb{C}^2$
and not directly in the complex manifold $\Delta$.
A similar approach is implicit in Hille's proof.
\begin{lemma} \label{single sequence}
Let the Cauchy problem \eqref{cauchy} be given.
Suppose that $\gamma: [0,1]\to\mathbb{C}$
is an arc starting at the initial point $z_0$ and that an analytic
continuation
$(R,\pi,j,\omega)$ of the initial solution $w$
can be carried out along $\gamma|_{[0,1)}$; let
$\widetilde\gamma :[0,1)\to R$ be the lifted arc of
$\gamma $ with respect to the natural projection $\pi $.
Consider the arc
$\theta:=\gamma \times [\omega \circ
\widetilde\gamma]:[0,1)\to\mathbb{C}^2$;
suppose that the initial known term $F\circ\eta$ can be analytically
continued along
$\theta$ and let
$\widetilde\theta:[0,1)\to\Delta$ be the lifted arc with respect to
the natural projection $p:\Delta\to\mathbb{C}^2$.
Finally, suppose that there exists a sequence $\{t_k\}\to 1$ such that
$\{(\widetilde\theta(t_k)\}$ converges to $\vartheta\in\Delta$ and that
the known term $F$ is holomorphic at $\vartheta$.
Then the initial solution $w$ admits an analytic continuation
along $\gamma$ up to the endpoint $\gamma(1)$.
\end{lemma}
\begin{proof}
Let $(z_k, w_k)$ be the coordinates of $p\circ\widetilde\theta(t_k)$ and
$(z_{\infty }, w_{\infty })$ those of $p(\vartheta )$.
Let $\eta_{\infty } $ be the branch of $p^{-1}$ such that
$\eta_{\infty }(\vartheta )=\vartheta $ and let
$$
F\circ\eta(z,w)= \sum_{r,s=0}^{\infty} c_{r,s}(z-z_k)^r(w-w_k)^s
$$
be the Taylor development of $F\circ\eta$ in a bidisc
$\mathbb{D}(z_{\infty }, w_{\infty}, \rho, \sigma)$
around $(z_{\infty }, w_{\infty })$.
The analytic continuation of $F\circ\eta$ along $\widetilde\theta$
can be concretely carried out
in the underlying $\mathbb{C}^2$ by a chain of bidiscs and Taylor
developments
$\{(\mathcal{U}_k, F\circ\eta_k )\}_{k\in\mathbb{N}}$, where
\begin{equation}
F\circ\eta_k (z,w)=
\sum_{r,s=0}^{\infty} c_{r,s,k}(z-z_k)^r(w-w_k)^s,\quad k\in\mathbb{N},
\label{taylor}
\end{equation}
$\gamma\times[\omega \circ \widetilde\gamma]
(t_k)\in \mathcal{U}_{N(k)}$ for every $k$ and some strictly increasing
function $N:\mathbb{N}\to\mathbb{N}$
(which we call the \emph{counting function})
and, for each $k$: $\eta_k$ is a local inverse of the projection
mapping $p$ and
$\eta_{N(k)}:\mathcal{U}_{N(k)}\to\Delta $ is the local inverse of $p$
such that $\eta_{N(k)}\circ p(\widetilde\theta (t_k))
=\widetilde\theta (t_k)$.
By continuity, $\{c_{r,s,k}\}\to c_{r,s}$ for all $r,s$
as $k\to\infty $, hence we can find $a>0$ and $b>0$
such that the developments in \eqref{taylor} converge absolutely
and uniformly in the closed bidiscs
$\overline{\mathbb{D}(u_k, v_k, a, b)}$.
By Cauchy estimates, this implies that there exists $T\in\mathbb{R}^+$ such that
$\sum_{r,s=0}^{\infty} | c_{r,s,k}| a^r b^s0$, $\varepsilon >0$ and
a finite subset $Q=\{\lambda_{\nu}\}\subset P$,
such that
$$
\{\omega \circ \widetilde\gamma (t_k)\}
\subset \overline{\mathbb{D}(0,r)}\setminus
\cup_{\lambda_{\nu} \in Q}
\mathbb{D}(\lambda_\nu, \varepsilon).
$$
Now, by continuity, there exists $\varrho >0$
such that
$$
z\in \mathbb{D}(\gamma (1), \varrho )\quad\text{implies}\quad
\mathop{\rm pr}\nolimits_{\mathbb{C}_w}
\big[\mathcal{S}\cap (\{z\}\times\mathbb{C}) \big]
\subset \cup_{\lambda_{\nu} \in Q}
\mathbb{D}(\lambda_\nu, \varepsilon);
$$
Set
$$
W:=\overline{\mathbb{D}(\gamma (1), \varrho )}
\times \Big[\overline{\mathbb{D}(0,r)}
\setminus \cup_{\lambda_{\nu} \in Q}
\mathbb{D}(\lambda_\nu, \varepsilon) \Big];
$$
by construction $W$ is compact in $\mathbb{C}^2$ and
$W\cap\mathcal{S}=\emptyset$;
also, we may suppose, without loss of generality,
$\{\gamma (t_k)\}\subset \mathbb{D}(\gamma (1), \varrho )$,
implying in turn that
\begin{equation}
\{\gamma \times [\omega \circ \widetilde\gamma](t_k) \}
\subset W.
\label{dentw}
\end{equation}
Let now $A$ be the holomorphic function on $\mathbb{C}^2$
such that $\mathcal{S}=A^{-1}(0)$;
the set
\begin{equation}\label{discrete}
\mathcal{B}:= \{\zeta \in R:A(\pi(\zeta), {\omega }(\zeta ) )=0
\}
\end{equation}
is discrete for otherwise we would have
$A(\pi(\zeta), {\omega }(\zeta ) )\equiv 0$ for
all $\zeta \in R$
contradicting the hypothesis that $(z_0,w_0)\not\in \mathcal{S}$.
Thus, by continuity. $\widetilde\gamma^{-1}(\mathcal{B})$ is discrete and,
by \eqref{dentw}, $\widetilde\gamma^{-1}(\mathcal{B})\cap\{t_k\}$
is finite.
Therefore, by passing to a nearby homotopic arc
if needed, we may suppose
$\widetilde\gamma([0,1))\cap\mathcal{B}=\emptyset $, implying
\[
\gamma \times[\omega \circ \widetilde\gamma]
([0,1))\cap\mathcal{S}=\emptyset.
\]
Hence $F\circ\eta $ admits regular analytic continuation along
$\theta := \gamma\times [\omega \circ\widetilde\gamma]
:[0,1)\to\mathbb{C}^2$.
Let $\widetilde\theta :[0,1)\to \Delta $
be the lifted arc with respect to the projection mapping
$p $.
By construction and by \eqref{dentw} we have
$$
\{\widetilde\theta(t_k)\}
\subset p^{-1}(\{\gamma \times
\omega \circ \widetilde\gamma (t_k))\}) \subset p^{-1}(W).
$$
Since $W$ is compact and $p$ is proper, $p^{-1}(W)$ is compact;
thus, by maybe passing to a subsequence, we may assume
that $\{\widetilde\theta(t_k)\}$
converges to a limit
$\vartheta \in p^{-1}(W)\subset \Delta \setminus \Sigma $;
note that $F$ is holomorphic at $\vartheta $; now a direct
application of lemma \ref{single sequence} allows us to conclude
that $w$ admits analytic continuation up to $\gamma(1)$.
This fact contradicts the hypothesis that
$\lim_{t\to 1}\omega\circ\widetilde\gamma(t)$
does not exist.
\end{proof}
\noindent
{\bf Remark} Theorem \ref{painleve'} implies that none of the
singularities of the solution of the Cauchy problem
\eqref{cauchy} are essential, for if $\zeta$ were such a singularity,
there would exists a path $\gamma:[0,1]\to\mathbb{C}$ connecting
$z_0$ to $\zeta$ such that $w$ admits analytic continuation along
$\gamma|_{[0,1)}$ and no continuous extension up to $\gamma(1)$.
\subsection*{Acknowledgements}
The author wishes to thank the anonymous referee for his or her valuable
remarks and criticisms, which helped us improve this article.
In particular, for suggesting the inclusion of counterexample \ref{noproper}.
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\end{document}