%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 182K, Plain TeX (version 3.141), 32 pages,
% 9 figures (automatically generated)
% for a postscript printer driven by dvips. The names of the
% figures are 1a.ps, 1b.ps, 2a.ps, 2b.ps, 2c.ps, 3a.ps, 3b.ps,
% 3c.ps, 4.ps.
% To print run through TeX and print the dvi file using dvips.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification = \magstep1
%
%The original version of these macros is due to J.P. Eckmann
%
%\magnification \magstep1
\vsize=22 truecm
\hsize=16 truecm
\hoffset=0.8 truecm
\normalbaselineskip=5.25mm
\baselineskip=5.25mm
\parskip=10pt
\immediate\openout1=key
\font\titlefont=cmbx10 scaled\magstep1
\font\authorfont=cmcsc10
\font\footfont=cmr7
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
\font\smaller=cmr5
%%%%%constant subscript positions%%%%%
\fontdimen16\tensy=2.7pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=2.7pt
%%%%%%%%%%%%%%%%%%%%%%
%%% macros %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\def\dowrite #1{\immediate\write16 {#1} \immediate\write1 {#1} }
%\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
\newcount\EQNcount \EQNcount=1
\newcount\SECTIONcount \SECTIONcount=0
\newcount\APPENDIXcount \APPENDIXcount=0
\newcount\CLAIMcount \CLAIMcount=1
\newcount\SUBSECTIONcount \SUBSECTIONcount=1
\def\SECTIONHEAD{X}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\QED{\hfill\smallskip
\line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
\ \ \ \ \ \ }
\bigskip}
% These ones cannot be used in amstex
% Make te symbols just bold
%\def\real{{\bf R}}
\def\rational{{\bf Q}}
%\def\natural{{\bf N}}
%\def\complex{{\bf C}}
%\def\integer{{\bf Z}}
\def\torus{{\bf T}}
% Make the symbols using kerning
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
$\scriptscriptstyle |$}}\kern-.40em{\rm C}}
%
% These ones can only be used in amstex
%
%\def\real{{\Bbb R}}
%\def\rational{{\Bbb Q}}
%\def\natural{{\Bbb N}}
%\def\complex{{\Bbb C}}
%\def\integer{{\Bbb Z}}
%\def\torus{{\Bbb T}}
%
%
%
\def\Re{{\rm Re\,}}
\def\Im{{\rm Im\,}}
\def\PROOF{\medskip\noindent{\bf Proof.\ }}
\def\REMARK{\medskip\noindent{\bf Remark.\ }}
\def\NOTATION{\medskip\noindent{\bf Notation.\ }}
\def\PRUEBA{\medskip\noindent{\bf Demostraci\'on.\ }}
\def\NOTA{\medskip\noindent{\bf Nota.\ }}
\def\NOTACION{\medskip\noindent{\bf Notaci\'on.\ }}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1 \dowrite{undefined equation #1}
\else\csname e#1\endcsname\fi}
\def\clm(#1){\ifundefined{c#1}$\clubsuit$#1 \dowrite{undefined claim #1}
\else\csname c#1\endcsname\fi}
\def\EQ(#1){\leqno\JPtag(#1)}
\def\NR(#1){&\JPtag(#1)\cr} %the same as &\tag(xx)\cr in eqalignno
\def\JPtag(#1){(\SECTIONHEAD.
\number\EQNcount)
\expandafter\xdef\csname
e#1\endcsname{(\SECTIONHEAD.\number\EQNcount)}
\dowrite{ EQ \equ(#1):#1 }
\global\advance\EQNcount by 1
}
\def\CLAIM #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~\SECTIONHEAD.\number\CLAIMcount.} {\sl #3}\par
\expandafter\xdef\csname c#2\endcsname{#1\
\SECTIONHEAD.\number\CLAIMcount}
%\immediate \write16{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
%\immediate \write1{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
\dowrite{ CLAIM #1 (\SECTIONHEAD.\number\CLAIMcount) :#2}
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\CLAIMNONR #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~#2} {\sl #3}\par
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\SECTION#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\def\SECTIONHEAD{\number\SECTIONcount}
\immediate\dowrite{ SECTION \SECTIONHEAD:#1}\leftline
\rightline{{\sectionfont \SECTIONHEAD.}\ {\sectionfont #1}}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\APPENDIX#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\def\SECTIONHEAD{\ifcase \number\APPENDIXcount X\or A\or B\or C\or D\or E\or F \fi}
\global\advance\APPENDIXcount by 1
\vfill \eject
\immediate\dowrite{ APPENDIX \SECTIONHEAD:#1}\leftline
{\titlefont APPENDIX \SECTIONHEAD: }
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SECTIONNONR#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\immediate\dowrite{SECTION:#1}\leftline
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SUBSECTION#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\def\SUBSECTIONHEAD{\number\SUBSECTIONcount}
\immediate\dowrite{ SUBSECTION \SECTIONHEAD.\SUBSECTIONHEAD :#1}\leftline
{\subsectionfont
\SECTIONHEAD.\number\SUBSECTIONcount.\ #1}
\global\advance\SUBSECTIONcount by 1
\nobreak\smallskip\noindent}
\def\SUBSECTIONNONR#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\immediate\dowrite{SUBSECTION:#1}\leftline{\subsectionfont
#1}
\nobreak\smallskip\noindent}
%%%%%%%%%%%%%TITLE PAGE%%%%%%%%%%%%%%%%%%%%
\let\endarg=\par
\def\finish{\def\endarg{\par\endgroup}}
\def\start{\endarg\begingroup}
\def\getNORMAL#1{{#1}}
\def\TITLE{\beginTITLE\getTITLE}
\def\beginTITLE{\start
\titlefont\baselineskip=1.728
\normalbaselineskip\rightskip=0pt plus1fil
\noindent
\def\endarg{\par\vskip.35in\endgroup}}
\def\getTITLE{\getNORMAL}
\def\AUTHOR{\beginAUTHOR\getAUTHOR}
\def\beginAUTHOR{\start
\vskip .25in\rm\noindent\finish}
\def\getAUTHOR{\getNORMAL}
\def\FROM{\beginFROM\getFROM}
\def\beginFROM{\start\baselineskip=3.0mm\normalbaselineskip=3.0mm
\obeylines\sl\finish}
\def\getFROM{\getNORMAL}
\def\ENDTITLE{\endarg}
\def\ABSTRACT#1\par{
\vskip 1in {\noindent\sectionfont Abstract.} #1 \par}
\def\ENDABSTRACT{\vfill\break}
\def\TODAY{\number\day~\ifcase\month\or January \or February \or March \or
April \or May \or June
\or July \or August \or September \or October \or November \or December \fi
\number\year}
\newcount\timecount
\timecount=\number\time
\divide\timecount by 60
\def\DRAFT{\font\footfont=cmti7
\footline={{\footfont \hfil File:\jobname, \TODAY, \number\timecount h}}
}
%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\period{\unskip.\spacefactor3000 { }}
%
% ...invisible stuff
%
\newbox\noboxJPE
\newbox\byboxJPE
\newbox\paperboxJPE
\newbox\yrboxJPE
\newbox\jourboxJPE
\newbox\pagesboxJPE
\newbox\volboxJPE
\newbox\preprintboxJPE
\newbox\toappearboxJPE
\newbox\bookboxJPE
\newbox\bybookboxJPE
\newbox\publisherboxJPE
\def\refclearJPE{
\setbox\noboxJPE=\null \gdef\isnoJPE{F}
\setbox\byboxJPE=\null \gdef\isbyJPE{F}
\setbox\paperboxJPE=\null \gdef\ispaperJPE{F}
\setbox\yrboxJPE=\null \gdef\isyrJPE{F}
\setbox\jourboxJPE=\null \gdef\isjourJPE{F}
\setbox\pagesboxJPE=\null \gdef\ispagesJPE{F}
\setbox\volboxJPE=\null \gdef\isvolJPE{F}
\setbox\preprintboxJPE=\null \gdef\ispreprintJPE{F}
\setbox\toappearboxJPE=\null \gdef\istoappearJPE{F}
\setbox\bookboxJPE=\null \gdef\isbookJPE{F} \gdef\isinbookJPE{F}
\setbox\bybookboxJPE=\null \gdef\isbybookJPE{F}
\setbox\publisherboxJPE=\null \gdef\ispublisherJPE{F}
}
\def\ref{\refclearJPE\bgroup}
\def\no {\egroup\gdef\isnoJPE{T}\setbox\noboxJPE=\hbox\bgroup}
\def\by {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
\def\paper{\egroup\gdef\ispaperJPE{T}\setbox\paperboxJPE=\hbox\bgroup}
\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
\def\vol{\egroup\gdef\isvolJPE{T}\setbox\volboxJPE=\hbox\bgroup\bf}
\def\preprint{\egroup\gdef
\ispreprintJPE{T}\setbox\preprintboxJPE=\hbox\bgroup}
\def\toappear{\egroup\gdef
\istoappearJPE{T}\setbox\toappearboxJPE=\hbox\bgroup}
\def\book{\egroup\gdef\isbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\publisher{\egroup\gdef
\ispublisherJPE{T}\setbox\publisherboxJPE=\hbox\bgroup}
\def\inbook{\egroup\gdef\isinbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\bybook{\egroup\gdef\isbybookJPE{T}\setbox\bybookboxJPE=\hbox\bgroup}
\def\endref{\egroup \sfcode`.=1000
\if T\isnoJPE \item{[\unhbox\noboxJPE\unskip]}
\else \item{} \fi
\if T\isbyJPE \unhbox\byboxJPE\unskip: \fi
\if T\ispaperJPE \unhbox\paperboxJPE\unskip\period \fi
\if T\isbookJPE ``\unhbox\bookboxJPE\unskip''\if T\ispublisherJPE, \else.
\fi\fi
\if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE,
\else\period \fi\fi
\if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi
\if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if
T\isyrJPE \ \else\period \fi\fi\fi
\if T\istoappearJPE (To appear)\period \fi
\if T\ispreprintJPE Preprint\period \fi
\if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi
\if T\isvolJPE \unhbox\volboxJPE\unskip, \fi
\if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi
\if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def \cite#1{{[#1]}}
%*************** TO GET SMALLER FONT FAMILIES *****************
\newskip\ttglue
% ********** EIGHT POINT **************
\def\eightpoint{\def\rm{\fam0\eightrm}
\textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\eightit \def\it{\fam\itfam\eightit}
\textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}
\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}
\textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}
\tt \ttglue=.5em plus.25em minus.15em
\normalbaselineskip=9pt
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}
\let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm}
\font\eightrm=cmr8 \font\sixrm=cmr6 \font\fiverm=cmr5
\font\eighti=cmmi8 \font\sixi=cmmi6 \font\fivei=cmmi5
\font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\fivesy=cmsy5
\font\eightit=cmti8 \font\eightsl=cmsl8 \font\eighttt=cmtt8
\font\eightbf=cmbx8 \font\sixbf=cmbx6 \font\fivebf=cmbx5
\def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy
\left#1\vbox to6.5pt{}\right.\enspace$}}}
%************** NINE POINT *****************
\def\ninepoint{\def\rm{\fam0\ninerm}
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\nineit \def\it{\fam\itfam\nineit}
\textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}
\textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett}
\textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}
\tt \ttglue=.5em plus.25em minus.15em
\normalbaselineskip=11pt
\setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}
\let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm}
\font\ninerm=cmr9 \font\sixrm=cmr6 \font\fiverm=cmr5
\font\ninei=cmmi9 \font\sixi=cmmi6 \font\fivei=cmmi5
\font\ninesy=cmsy9 \font\sixsy=cmsy6 \font\fivesy=cmsy5
\font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninett=cmtt9
\font\ninebf=cmbx9 \font\sixbf=cmbx6 \font\fivebf=cmbx5
\def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy
\left#1\vbox to7.25pt{}\right.$}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%************* The picture definitions ******************
\def\picture#1#2{
\vbox to 2.8 in{
\hbox to 2 in
{\special{psfile = #1.ps angle = 270 hscale = 40 vscale = 40 }}
\hfil \vfil}
\kern1.3cm
\vbox{\eightpoint #2 \hfil}
\vskip0.3truecm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def \cite#1{{[#1]}}
\def\cite#1{{\rm [#1]}}
\def\bref#1{{\rm [~\enspace~]}} % blank ref cite
\def\degree{\mathop{\rm degree}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\bT{{\bf T}}
\def\frac#1#2{{#1\over#2}}
\def\Norm{{\|~\enspace~\|}}
\TITLE
On the singularity structure of invariant curves of symplectic mappings.
\footnote{$^1$}
{\baselineskip=12pt\rm Preprint available
through the mathematical physics electronic
preprint archive. Send mail to {\tt mp\_arc@math.utexas.edu} for details.}
\ENDTITLE
\AUTHOR
Rafael de la Llave
\footnote{$^2$}{e-mail: {\tt llave@math.utexas.edu}}
\footnote{$^3$}{Supported in part by NSF Grant DMS 9201143}
\footnote{$^4$}{Supported in part by TARP 3658-014}
{\rm and} Stathis Tompaidis
\footnote{$^5$}{e-mail: {\tt stathis@math.utexas.edu}}
$^3$ $^4$
\footnote{$^6$}{Address after September 1 1994: Dept. of Mathematics,
University of Toronto, 100 St. George Street, Toronto, ON M5S 1A1, Canada}
\FROM
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712-1082
\ENDTITLE
\ABSTRACT
We study invariant curves in standard-like maps conjugate to rigid rotation
with complex frequencies.
The main goal is to
study the analyticity domain of the functions
defined perturbatively by Lindstedt perturbation
expansions. We argue, based on infinite-dimensional bifurcation
theory that the boundary of analyticity should typically
consist of branch points of order two and we verify it in
some examples using non-perturbative numerical methods.
We show that this nature of the singularities of the analyticity domain can
explain previously reported numerical results and also
suggests other numerical methods.
\ENDABSTRACT
\SECTION Introduction.
Invariant curves have been widely studied as an important landmark
that organizes the long term behavior. Notably, for two dimensional systems,
they completely prevent long term diffusion and,
for systems of more dimensions, even if not preventing
diffusion completely, are still the main obstacles.
The standard way of computing invariant curves is to introduce a
parametrization that identifies curves with functions and then study
the functional equation that expresses that the curve is invariant
and that the motion on it is a rigid rotation in some distorted coordinates.
The amount of rotation -- henceforth called the {\sl frequency} of
the curve -- appears as a parameter in the functional equation and can
be used to label the curves.
For families of maps that contain an integrable -- i.e. explicitly
solvable in closed form -- system, one can study the functional
equation for invariant curves perturbatively and one is led to the
Lindstedt expansions of classical mechanics (see \cite{Po} and
section 3).
Even if these expansions have been in use for over a century, their
analytic properties have been very hard to study. For example, due to
the presence of {\sl small divisors} for diophantine frequencies
(see section 3), the fact that
they have a positive radius of convergence was established only in
the late 50's with K.~A.~M. theory (see e.g. \cite{SM}) and numerical
values with the right order of magnitude have only been achieved in
the 80's with the use of {\sl computer assisted proofs} (see \cite{R},
\cite{LR},\cite{CC}).
The goal of this paper is to study the domain of analyticity of these
series and to study the nature of the singularities at the boundary
(previous papers concerned with the numerical computation of the
domain of analyticity of invariant curves
have been: \cite{BC}, \cite{BCCF}, \cite{FL}, \cite{BM} \cite{LT1}, \cite{LT2},
\cite{BT}, \cite{AB}, \cite{BMT}).
Notice that once that we know qualitatively what is the nature of the
singularities at the boundary it is possible to devise interpolation
and extrapolation schemes that, being well adapted to the functions
we are dealing with, are quite efficient.
Following \cite{BM} and \cite{BT} we will regularize the problem
by considering complex values of the frequency. The regularized
equations, as we will see, have no small divisors and one can argue
what is the structure of the boundary of analyticity. These insights
-- theoretical and numerical -- can be transferred to the physical
case of purely real frequencies by taking the limit as the imaginary
part of the frequency tends to zero.
Our first result is a rigorous theorem that states that, for complex
frequencies with non-zero imaginary part, provided that certain
non-degeneracy conditions hold, the nature of the singularities is
discrete branch points of order two. The theorem is based in the
theory of bifurcation of compact operators and the non-degeneracy
conditions -- which hold generically --
can be checked \`a posteriori by a finite calculation.
To verify the applicability of the theorem, we implement a
non-perturbative continuation method and indeed confirm that for the
most standard examples, the boundary of analyticity consists of
branch points of order two.
Finally, we discuss the implication of these results for the
applicability of analytic extrapolation methods. By far, the most
commonly used method for analytic extrapolation is the use of Pad\'e approximants.
For Lindstedt series, extrapolation Pad\'e
methods were used in \cite{BC} and used and
refined in \cite{BCCF}, \cite{LT1}. For Lindstedt series with complex frequencies
they were used in \cite{BM}. Since the Pad\'e method is based on
rational approximation, it is rather delicate to use for functions
whose singularities are branch points and not just poles.
The behavior of Pad\'e approximants for functions with branch points
has been considered in \cite{N1}, \cite{N2}, \cite{HB1}, \cite{HB2}
which make very precise
conjectures about their behavior and support them with analytical and
numerical evidence.
We observe that the numerical results of \cite{BM} -- which we
reproduce and confirm -- can be explained and fit very well the
conjectures of \cite{N1} and \cite{N2} for functions with the singularity structure
obtained by our non-perturbative results.
Moreover, the knowledge of the nature of the singularities can be used
to devise methods that are better adapted than the Pad\'e rational
extrapolations. In our last section, we introduce the so-called
{\sl logarithmic Pad\'e} method \cite{BGM} and we discuss the results
of implementing it as well as the {\sl ratio} method.
All the numerical results confirm spectacularly well that the
singularities of the examples we consider are branch points of order
two and that, as the real part of the frequency approaches zero,
these branch points accumulate to the natural boundary that has
been previously reported in the literature. The details of this
accumulation, remain as a challenge. We also hope that the
approximation of the natural boundaries by branch points may
also be useful in other problems related to K.~A.~M. theory.
\SECTION Definitions and Notation.
We will study topologically nontrivial circles,
invariant under an area-preserving map belonging in an
one-parameter family of maps.
The maps, henceforth called ``standard like'' are given by
$$F_{\epsilon} (q, p)
= (q + p + \epsilon S(q) \bmod 2\pi,\ p + \epsilon S(q))
\EQ(def)$$
The variables $p,q$ and the parameter $\epsilon$ will be considered complex
and the function $S$ is $2 \pi$ periodic, analytic, with zero average over
$[-\pi, \pi]$.
The case $S(q) = \sin(q)$, called the standard map, has been
extensively studied as a qualitative model of several physical phenomena
(see \cite{A}, \cite{Ch}) and as a typical example of breakdown of invariant
curves (See \cite{Gr}, \cite{McK}). We will also study the case
when $S(q)$ is an odd trigonometric polynomial that can be considered as an
extension of the usual standard map.
We define the frequency
$\omega = \lim_{n \to \infty} \pi_1 \tilde F^n (q,p) /n\quad \mod 2 \pi$,
whenever the limit exists, with $\tilde F$ a lift of $F$ and $\pi_1$
the projection to the first coordinate $\pi_1 (q,p) = q$.
Notice that for $\epsilon = 0$ the map $F_0$ has an invariant curve for
every complex frequency $\omega$ with $p = \omega$.
We will identify invariant curves by the fixed frequency $\omega$.
As the parameter $\epsilon$ varies an invariant curve is distorted and can
even disappear (as for example an invariant curve
with rational frequency for $\epsilon \ne 0$).
\SECTION Lindstedt perturbation expansions
If an invariant curve is conjugate to a rigid rotation with frequency
$\omega$,then it can be parameterized by
two functions $p$ and $q$ of a variable $\theta$ which satisfy:
$$F_\epsilon (q(\theta), p(\theta)) = (q(\theta + \omega), p(\theta + \omega)).
\EQ(conj)$$
Moreover, assuming the invariant curve is a graph
$$ q(\theta) = \theta + u_\epsilon (\theta), \quad
p(\theta) = \omega + u_\epsilon(\theta) - u_\epsilon(\theta - \omega)$$
The conjugating function $u_\epsilon$ is called the
hull function (see \cite{A}). Combining \equ(def), \equ(conj)
$$ \Delta_\omega u_\epsilon (\theta) = \epsilon S(u_\epsilon (\theta) + \theta)
\EQ(stdmap)$$
where $\Delta_\omega$ is the operator defined by
$$\Delta_\omega u_\epsilon (\theta) = u_\epsilon(\theta + \omega) -
2 u_\epsilon (\theta) + u_\epsilon (\theta - \omega).
\EQ(deltaomega)$$
Notice that a solution of \equ(stdmap) corresponds to a solution of
\equ(conj) but the
opposite need not be the case (for the case of real variables the
Birkhoff theorem (\cite{M}, \cite{F})
guarantees that the conjugating function is a graph but
we are not aware of a similar result for the case of complex variables).
If there is a solution $u_\epsilon$, for fixed $\epsilon$, to \equ(stdmap)
then $u_\epsilon (\theta + \alpha) - \alpha$, for $\alpha \in \complex$,
is also a solution. To fix this ambiguity we choose the normalization
$$\int_{-\pi}^{\pi} u_\epsilon (\theta) d \theta = 0.
\EQ(normalization)$$
Under this normalization $u_0 = 0$ is the solution for $\epsilon = 0$.
To compute the unknown hull function $u_\epsilon$
we assume that it can be expanded in a power series in
$\epsilon$
$$u_\epsilon (\theta) = \sum_{n=0}^\infty \epsilon^n u_n (\theta).
\EQ(lindstedt)$$
The series \equ(lindstedt) are called Lindstedt series. Substituting in
\equ(stdmap), expanding and matching formally corresponding
orders in $\epsilon$
$$\eqalign{
u_0 (\theta) & = 0\cr
\Delta_\omega u_1 (\theta) & = S(\theta) \cr
\Delta_\omega u_n (\theta) & = R_n(\theta) = \frac{1}{(n-1)!}
\frac{d^{(n-1)}}{d\epsilon^{(n-1)}} \bigl |_{\epsilon = 0}
S(\theta + \sum_{l = 1}^{\infty} u_l(\theta) \epsilon^l), \quad n \ge 2\cr
}
\EQ(formal)$$
Notice that the right hand side of \equ(formal) depends only on
$u_1, \dots, u_{n-1}$ so that one can solve recursively, provided that
$$\Delta_\omega u_n (\theta) = R_n(\theta)$$
has a solution, i.e.
$$\int_{-\pi}^{\pi} R_n(\theta) d\theta = 0.
\EQ(compatibility)$$
>From the normalization condition on the hull function \equ(normalization)
we have
$$\int_{-\pi}^{\pi} u_n (\theta) d \theta = 0,\qquad n \ge 1 $$
and, using the properties of $S$ we can show inductively that the
compatibility condition \equ(compatibility) is satisfied (see also
\cite{FL}).
In effect, if we rewrite $R_n$ as
$$\eqalign{
R_n (\theta) & = \frac{1}{(n-1)!}
\frac{d^{(n-1)}}{d\epsilon^{(n-1)}} \bigl |_{\epsilon = 0} \left \{
\sum_{m=0}^{\infty} \frac{1}{m!} \frac{d^m}{d\theta^m} S(\theta)
\left ( \sum_{l=1}^\infty u_l (\theta) \epsilon^l \right )^m \right \} \cr
& = \sum_{m=0}^{\infty} ( \frac{1}{m!} \frac{d^m}{d\theta^m} S(\theta) )
\frac{1}{(n-1)!} \frac{d^{(n-1)}}{d\epsilon^{(n-1)}} \bigl |_{\epsilon = 0}
\left ( \sum_{l=1}^\infty u_l (\theta) \epsilon^l \right )^m \cr
& = \sum_{j=0}^{n-1} ( \frac{1}{j!} \frac{d^j}{d\theta^j} S(\theta) )
\sum_{n_1 + \cdots + n_j = n-1} \frac{n_1! \dots n_j!}{(n-1)!}
u_{n_1} (\theta) \cdots u_{n_j} (\theta), \ n\ge 2. \cr
}
\EQ(formal1)$$
In terms of Fourier series, the operator $\Delta_\omega$
is diagonal and, if
$$
u_n (\theta) = \sum_{k\ne 0} \hat u_{n,k} e^{ ik\theta}, \quad
R_n(\theta) = \sum_{k\ne 0} \hat R_{n,k} e^{ ik\theta}
$$
then
$$\hat u_{n,k} = \frac{\hat R_{n,k}}{2(\cos( k\omega ) -1)},\quad
n \ge 1,\quad k \ne 0.
$$
For the case of $\omega $ real, diophantine it was shown (see \cite{SM}
for an analytic proof,
\cite{CC} for a computer-aided one)
using methods from KAM theory, that the Lindstedt series for the hull
function, converges to
an analytic function for $| \epsilon | < \rho$ for some $\rho > 0$.
When $\Im \omega \ne 0$, $2(\cos( k\omega ) -1)$ is bounded away
from zero uniformly in $k \in \integer - \{ 0\} $ so that
$$\sup_{k \ne 0} |2(\cos( k\omega ) -1)|^{-1} \le K_\omega.
\EQ(bound)$$
To estimate convergence of the Lindstedt series we introduce the norm
$\| \ \|_\delta$ with
$\|f\|_\delta = \sup_{k\in \integer} e^{\delta |k|} |\hat f_k|$, where
$\hat f_k$ is the
$k^{\rm th}$ Fourier coefficient of $f$. This defines a norm on a Banach
space of analytic functions denoted henceforth as $C^{\omega,\delta}$.
Since $S$ is a $2\pi$ periodic, analytic function, using Cauchy
inequalities we can find a constant $K_\delta ( = 2/\delta)$ such that
$\| \frac{1}{j!} \frac{d^j}{d\theta^j} S(\theta) \|_\delta \le K_\delta^j \sup_{\theta \in {\rm I}_{2 \delta}} | S(\theta) |$,
where ${\rm I}_{2 \delta} \equiv \{ \theta : |\Im \theta | < 2\delta\}$.
\CLAIM {Theorem}(convergence)
For $|\epsilon |$ sufficiently small,
the Lindstedt series \equ(lindstedt)
converges uniformly to an analytic function defined on
${\rm I}_\delta \equiv \{\theta : | \Im \theta| < \delta\}$.
\PROOF
>From \equ(formal) we have $\| u_n \|_\delta \le K_\omega \| R_n \|_\delta$.
For $n=1, R_1 (\theta) = S(\theta)$ and
$\|u_1\|_\delta \le K_\omega \sup_{\theta \in {\rm I}_{2 \delta}} |S(\theta)|$.
For $n > 1$, from \equ(formal1),
$$\|u_n \|_\delta \le K_\omega \|R_n\|_\delta
\le K_\omega
\sum_{j=1}^{n-1} \| \frac{1}{j!} \frac{d^j}{d\theta^j} S(\theta) \|_\delta
\sum_{n_1 + \cdots + n_j = n-1}
\frac{n_1! \dots n_j!}{(n-1)!} \|u_{n_1}\|_\delta \dots \|u_{n_j}\|_\delta.
\EQ(bound2)$$
To estimate the size of $\|u_n\|_\delta$, we introduce the function
$\phi : \real \to \real$ with
$\phi (z) = \sum_{n=0}^\infty \frac{1}{n!} \|\frac{d^n}{d\theta^n} S(\theta) \|_\delta z^n$.
Since
$\frac{1}{n!} \|\frac{d^n}{d\theta^n} S(\theta) \|_\delta \le K_\delta^n \sup_{\theta \in {\rm I}_{2 \delta}} | S(\theta) |$,
$\phi$ is an analytic function for $|z| < K_\delta^{-1}$. We can bound
$\|u_n\|_\delta$ by the coefficients $\sigma_n$ of
$\sigma(z) = \sum_{n=0}^{\infty} \sigma_n z^n$, where
$$\sigma(z) = K_\omega z \phi(\sigma(z)), \quad \sigma (0) = 0.
\EQ(sigma)$$
By induction we verify that $\|u_n\|_\delta \le \sigma_n, \ n\ge 1$.
Moreover, since $\phi$ is an analytic function and
$\|S(\theta)\|_\delta \ne 0$, by the implicit function theorem,
$\sigma$ is analytic for $|z|$ small enough and we can bound
$\sigma_n \le \alpha^n$, for some $\alpha >0$.
This implies that the Lindstedt series converges uniformly to an analytic
function in $I_\delta$, for
$\epsilon < 1 / \alpha $,
and concludes the proof of \clm(convergence).
\QED
\def\calT{{\cal T}}
\SECTION Bifurcation from a simple eigenvalue
We will analyze the equation satisfied by the hull function \equ(stdmap)
using methods from functional analysis and bifurcation theory. For fixed
$\omega$ we define the operator
$\calT : \complex \times C_0^0 \to C_0^0$
$$\calT (\epsilon, u) (\theta) = \epsilon \Delta_\omega^{-1}
S(u(\theta) + \theta)
\EQ(tau)$$
where $C_0^0$ is the space of $2\pi$ periodic, continuous, complex functions
with zero average on $[-\pi, \pi]$ under the supremum norm, i.e.
$\|f\|_{C_0^0} = \max_{\theta \in [-\pi, \pi]} |f(\theta)|$. It is
well known that under this norm $C_0^0$ is a Banach space.
Also, since $\complex$ is a Banach space, $\complex \times C_0^0$ is a
Banach space under the norm
$\|(\epsilon, u)\|_{\complex \times C_0^0} = |\epsilon | + \| u\|_{C_0^0}$.
\CLAIM{Lemma}(deltainverse)
The operator $\Delta_\omega : C_0^0 \to C_0^0$, given
by \equ(deltaomega) is invertible
and, for $\Im \omega \ne 0$ the inverse is bounded.
\PROOF
Decomposing $\Delta_\omega$ in Fourier coefficients we have
$$(\Delta_\omega)_{k,l} = (e^{i\omega k} + e^{-i\omega k} -2)\delta_{k,l},
\qquad k\ne 0.$$
Since $(\Delta_\omega)_{k,l}$ is a diagonal matrix we can verify that
the inverse
$\Delta_\omega^{-1} : C_0^0 \to C_0^0$ exists and is given by
$[\Delta_\omega^{-1}]_{k,l} = [(\Delta_\omega)_{k,k}]^{-1} \delta_{k,l}, \quad k \ne 0$
(notice that zero average is essential for the existence of the inverse).
To show that $\Delta_\omega^{-1}$ is bounded for $\Im \omega \ne 0$, let
$\eta \in C_0^0, \| \eta \|_{C_0^0} \le M$. Then, decomposing in Fourier
coefficients
$$\eqalign{
\|\Delta_\omega^{-1} \eta \|_{C_0^0} & = \max_{\theta \in [-\pi, \pi]}
\bigl | \sum_{k\ne 0} \frac{\eta_k e^{ik\theta}}{e^{ik\omega} + e^{-ik\omega}-2}
\bigr | \cr
& \le \| \eta \|_{C_0^0} \sum_{k\ne 0}
\bigl | \frac{1}{e^{ik\omega} + e^{-ik\omega}-2} \bigr | \cr
& = 2 \| \eta \|_{C_0^0} \sum_{k > 0} \bigl |
\frac{1}{e^{ik\omega} + e^{-ik\omega}-2} \bigr | \cr}
$$
The sum is finite for $\Im \omega \ne 0$, since the denominator grows
asymptotically like
$e^{k | \Im \omega|}$. This concludes the proof.
\QED
We will now show some interesting properties of $\calT$.
\CLAIM{Lemma}(frechet)
The operator $\calT$, defined by \equ(tau), is
Fr\'echet differentiable for all $(\epsilon, u) \in \complex \times C_0^0$
with
$$D\calT (\epsilon, u) (\zeta, \eta) =
\zeta \Delta_\omega^{-1} S(u+\theta) + \epsilon \Delta_\omega^{-1}
[S'(u + \theta) \eta].$$
\PROOF
We have the following simple computation
$$\eqalign{
\| \calT (\epsilon + \zeta, u + \eta) - \calT (\epsilon, u) - &
D\calT (\epsilon, u) (\zeta, \eta) \|_{C_0^0} = \cr
& = \| (\epsilon + \zeta) \Delta_\omega^{-1} [S(u+\theta + \eta)
-S(u + \theta) - S'(u + \theta) \eta] \|_{C_0^0} \cr
& = \| (\epsilon + \zeta) \Delta_\omega^{-1} [S'' (\tilde u + \theta)
(\eta)^2 ] \|_{C_0^0} \cr
& \le |\epsilon + \zeta | \| \Delta_\omega^{-1} \|_{C_0^0}
\|S''\|_{C_0^0} \| \eta \|_{C_0^0}^2 \cr
}$$
\QED
\clm(frechet) implies that $\calT$ is complex analytic. We follow
\cite{ChH, pg. 23} in our definition of a complex analytic operator.
\CLAIM{Definition}(complexanal)
Let $X,Y$ be Banach spaces over $\complex$ and $U$ be a connected
open set of $X$. A function $f : X \to Y$ is complex analytic in
$U$ if, for each $x\in U$, there is a $\delta (x, h) > 0$, such that,
for each $y^* \in Y^*$, $f(x)$ is
single valued and $$ is an analytic function of
$t$ for $|t| < \delta(x,h)$.
We state the following theorem, based on the \clm(complexanal).
\CLAIM{Theorem}(analyticity)
(\cite{ChH, pg. 23, theorem 1.10})\
If $U$ is an open connected set of $X$, $f:U\to Y$ is
single valued and locally bounded, then the following statements are
equivalent
\leftline{(i) $f$ is complex analytic in $U$}
\leftline{(ii) $f$ is Fr\'echet differentiable in $U$}
\leftline{(iii) $f$ has infinitely many Fr\'echet derivatives}
\CLAIM{Theorem}(compactness)
For fixed $\epsilon_0$, $u_0$
and $\omega$, with $\Im \omega \ne 0$, the Fr\'echet
derivative of the map $u \to \calT (\epsilon_0, u)$, at $u = u_0$ is a
compact operator on $C_0^0$.
\PROOF
The Fr\'echet derivative of
$u \to \calT (\epsilon_0, u)$ exists and is given by
$$D_u \calT (\epsilon_0, u_0) \eta = \epsilon_0 \Delta_\omega^{-1}
[S' (u_0 + \theta) \eta].$$
To show that $D_u \calT (\epsilon_0, u_0)$ is a compact operator, we have to
show that it maps bounded sets to precompact ones.
Notice that $D_u \calT (\epsilon_0, u_0)$ is bounded, i.e. for
$\eta$ such that $\| \eta \|_{C_0^0} \le M$
$$\|D_u \calT (\epsilon_0, u_0)\eta \|_{C_0^0} \le |\epsilon_0 |
\| \Delta_\omega^{-1} \|_{C_0^0} \max_{\theta \in [-\pi, \pi]} |S'(\theta)|
\|\eta \|_{C_0^0}.$$
We will show that
$\{D_u \calT (\epsilon_0, u_0)\eta : \| \eta \|_{C_0^0} \le M\}$
is equicontinuous. We have
$$\eqalign{
|D_u \calT (\epsilon_0, u_0)\eta (\theta ') -
D_u \calT (\epsilon_0, u_0) \eta (\theta) & | = \cr
& \bigl |\epsilon_0
\Delta_\omega^{-1} [ S'(u_0(\theta ') + \theta ' ) \eta (\theta ') -
S'(u_0(\theta) + \theta) \eta(\theta) ] \bigr | \cr
& = |\epsilon_0 | \bigl | \Delta_\omega^{-1} [F(\theta ') - F(\theta)] \bigr |
\cr}$$
where $F(\theta) = S'(u(\theta) + \theta) \eta(\theta)$.
Decomposing into Fourier coefficients,
$$\eqalign{
| \Delta_\omega^{-1} [F(\theta ') - F(\theta)] | & =
\bigl | \sum_{k\ne 0} \frac{F_k (e^{ik\theta '} - e^{ik\theta} )}{e^{ik\omega}
+ e^{-ik\omega} -2} \bigl | \cr
& \le \|F\|_{C_0^0} \sum_{k\ne 0}
\frac{|k| |\theta - \theta '| \max{\theta ''
\in [-\pi , \pi]} |e^{ik\theta ''} |}{|e^{ik\omega}+ e^{-ik\omega} -2|} \cr
& \le \|S' \eta \|_{C_0^0} |\theta ' - \theta | \sum_{k\ne 0}
\frac{|k|}{|e^{ik\omega}+ e^{-ik\omega} -2|} \cr
}$$
As in the proof of \clm(deltainverse) the sum is finite for $\Im \omega \ne 0$.
By the Ascoli theorem (see \cite{Rudin, pg. 394, theorem A5}) we
conclude that $D_u \calT (\epsilon_0, u_0)$ is a compact operator.
\QED
Since $D_u \calT (\epsilon_0, u_0)$ is a compact operator the spectrum
of $D_u \calT (\epsilon_0, u_0)$ is discrete and has no accumulation points,
apart from zero (see \cite{Rudin}). Based on this property of
the spectrum we are able to characterize the nature of the singularities
in the analyticity domain of the hull function. This characterization
is based on the following theorem from bifurcation theory of operators
in Banach spaces.
\CLAIM{Theorem}(bifurcation)
For $B$ a Banach space over $\complex$ , let $K: \complex \times B \to B$ be a
Fr\'echet differentiable operator. Assume that $D_u K(0,0) = A$ (the
Fr\'echet derivative of the map $u \to K(0,u)$ at $u = 0$)
has a simple isolated
eigenvalue $0$, and
{\rm Null}($A$) = {\rm span}($v_0$), {\rm Null}($A^*$) = {\rm span}($w_0^*$),
for $v_0 \in B, w_0^* \in B^*$ and the co-dimension of ${\rm Range}(A) = 1$. If
$$ \ne 0, \qquad
\ne 0$$
then,
there are two distinct solutions to
$$K(\epsilon, u) = 0 \EQ(fixedpoint)$$
for
$0 < \epsilon < \rho$, for some $\rho > 0$, analytic in $\epsilon^{1/2}$.
\PROOF
The proof is divided in two parts. First we will use the Liapunov-Schmidt
reduction method to reduce \equ(fixedpoint) to an one dimensional scalar
equation. Then we will apply the Newton polygon method to deduce the form
of the solutions to the one dimensional equation (see \cite{GS}, \cite{ChH}
for examples and applications of these methods).
Since the null space of $A$ is one dimensional and the co-dimension of
Range($A$) is one, we can decompose $B$ into
$$ B = {\rm Null}(A) \oplus M,\qquad B = N \oplus {\rm Range}(A).$$
Since $B = N \oplus {\rm Range}(A)$, there exists (see
\cite{Rudin, pg. 133, theorem 5.16}) a continuous
projection $E : B \to {\rm Range}(A)$ with ${\rm Range}(E) = {\rm Range}(A)$,
${\rm Null}(E) = N$. Then, $K(\epsilon, u)=0$
is equivalent to
$$\eqalign{EK(\epsilon, u) = 0 \cr
(I-E)K(\epsilon, u) = 0 \cr
}\EQ(decomposition)$$
Following the Liapunov-Schmidt reduction method we will use the implicit
function theorem to solve the first of \equ(decomposition) and then
substitute the solution in the second. Thus we will finally have to
study a k-dimensional equation for a k-dimensional null space (we
assumed $k=1$).
Let $u = v+w, v\in {\rm Null}(A), w \in M$. Define the map
$F: M\times {\rm Null}(A) \times \complex \to {\rm Range}(A)$
$$ F(w, v, \epsilon) = EK(\epsilon, w+v) .
\EQ(projected)
$$
We have that $F(0,0,0) = 0, D_w F(0,0,0) = A$. Since $A: M \to {\rm Range}(A)$
is one to one and onto, and Range($A$) is closed, $A$ restricted to $M$
is invertible. From \clm(analyticity)
Fr\'echet differentiability is equivalent to complex analyticity.
By the implicit function
theorem the solution of \equ(projected) in a neighborhood of
$(0,0) \in {\rm Null}(A) \times \complex$ is a complex analytic function
$W: {\rm Null}(A) \times \complex \to M$ such that
$$F(W(v, \epsilon), v, \epsilon) = EK(\epsilon, v+W(v,\epsilon)) = 0 .$$
Substituting in the second of the equations \equ(decomposition)
$$(I-E)K(\epsilon, v + W(v, \epsilon)) = 0.
\EQ(reduced1)$$
>From \cite{Rudin pg. 99, theorem 4.12}
$${\rm Null}(A^*) = {\rm Range}(A)^\perp = \{ w^* \in B^* | = 0,
\forall u \in {\rm Range}(A) \}$$
We define $g:\complex \times \complex \to \complex$,
$$g(z,\epsilon) = =
.$$
>From the Fredholm alternative
$$g (z, \epsilon) = 0 \iff
(I-E)K(\epsilon, zv_0 + W(zv_0, \epsilon)) = 0$$
The function $g$ is analytic in a neighborhood of $(0,0)$ since
$K, W$ are complex analytic and we compute
$$\eqalign{
& g(0,0) = 0 \cr
& g_z(0,0) = 0 \cr
& g_{zz}(0,0) = \ne 0 \cr
& g_\epsilon(0,0) = \ne 0 \cr
& g(z, \epsilon) = \frac12 g_{zz} (0,0) z^2 + g_\epsilon(0,0) \epsilon +
O(\epsilon^2) + O(z^3) + O(\epsilon z) \cr}
$$
The Newton polygon is a method to identify the leading singularities
for solutions of $f(z,\epsilon) = 0$ where $f$ is a power series in $z$
whose coefficients can be power series in $\epsilon$. For $g(z, \epsilon)=0$,
the leading singular behavior is $\epsilon^{1/2}$ and if we perform the change
of variables $z = \epsilon^{1/2} y$ and divide $g(z(y), \epsilon)$
by $\epsilon$ we have
$$ \frac12 g_{zz}(0,0) y^2 + g_\epsilon(0,0) + O(\epsilon^{1/2}) = 0.
\EQ(newtonpolygon)$$
For $\epsilon = 0$ \equ(newtonpolygon) has two distinct solutions
$$y = \pm (-\frac{2 g_\epsilon(0,0)}{g_{zz}(0,0)} )^{1/2}
$$
By the implicit function theorem, there are two and only two solutions
$y_i(\epsilon), i = 1,2$, analytic in $\epsilon$, for $|\epsilon|$ sufficiently
small. This implies $z_i = \epsilon^{1/2} y_i(\epsilon)$ and, for
$\epsilon$ sufficiently small, there are two solutions to $K(u,\epsilon)=0$
analytic in $\epsilon^{1/2}$, with
$$u_i = W(\epsilon^{1/2} y_i(\epsilon)v_0, \epsilon) + \epsilon^{1/2}
y_i(\epsilon) v_0, \qquad i = 1,2.$$
This concludes the proof of the theorem.
\QED
\CLAIM{Corollary}(branch)
Suppose that for $(\epsilon_0, u_0)$ such that $\calT(\epsilon_0, u_0) = u_0$
(where $\calT$ defined in \equ(tau)), fixed $\omega$ with $\Im \omega \ne 0$,
$D_u \calT (\epsilon_0, u_0)$ has a simple eigenvalue 1. Let
$${\rm Null}(D_u \calT (\epsilon_0, u_0) - I) = {\rm span}(v_0),\quad
{\rm Null}\bigl [(D_u \calT (\epsilon_0, u_0) - I)^* \bigr ] =
{\rm span}(w_0^*).$$
If $ \ne 0$ and
$ \ne 0$
then,
$\epsilon_0$ is an isolated branch point of order 2, in the analyticity domain
of the hull function.
\PROOF
Consider the operator
$$K(\epsilon, u) = \calT (\epsilon + \epsilon_0, u + u_0) -I.$$
Then,
$$\eqalign{
& D_{uu}K(0,0) = D_{uu} \calT (\epsilon_0, u_0) \cr
& D_\epsilon K(0,0) = D_\epsilon \calT (\epsilon_0, u_0) \cr
}$$
>From \clm(frechet) $\calT$ is Fr\'echet differentiable. From
\clm(compactness) $D_u \calT(\epsilon_0, u_0)$ is a compact operator
and may exhibit isolated eigenvalues. An isolated simple eigenvalue
1 for $D_u \calT(\epsilon_0, u_0)$ corresponds to an isolated simple
eigenvalue 0 for $D_u \calT (\epsilon_0, u_0) - I$ with an
one dimensional null space and range of co-dimension 1. Moreover, since
$D_u \calT(\epsilon_0, u_0)$ is compact, we have
$B = {\rm Null}(D_u \calT (\epsilon_0, u_0) - I) \oplus {\rm Range}(D_u \calT (\epsilon_0, u_0) - I)$ (see \cite{Rudin}). Thus $K$ fulfills all the
conditions of \clm(bifurcation).
\QED
\REMARK The dual space of $C_0^0$ is the space of complex measures on
$[-\pi, \pi]$ with zero Fourier coefficient equal to zero. If $S$
is an odd trigonometric polynomial, then $u_0, v_0$ are odd functions
and the Fourier decomposition of the operator $D_u\calT (\epsilon_0, u_0)$
is the same
in $C_0^0$ and $(C_0^0)^*$. Thus
$$ \eqalign{
\quad =&
\int_{-\pi}^{\pi}
\epsilon_0 v_0(\theta) \Delta_\omega^{-1}
[S''(u_0(\theta) + \theta) v_0^2(\theta)] d\theta \cr
\quad =& \int_{-\pi}^{\pi}
v_0(\theta) \Delta_\omega^{-1} [S(u_0(\theta) + \theta)] d\theta}
\EQ(explicitnondeg)
$$
are integrals of even functions over an interval centered at the origin.
Thus, typically, the non-degeneracy conditions of \clm(branch) are
satisfied. This remark applies in particular to the case of the standard
map and it provides an explanation on the nature of the singularities of
the hull functions.
Notice that in the presence of an additional parameter one expects
cases where the non-degeneracy conditions are not satisfied and
different bifurcations may occur.
Notice that, because of the presence of the
compact operator $\Delta_\omega^{-1}$ in the
integrand in \equ(explicitnondeg), the high order Fourier coefficients in the
expansion of $u_0, v_0$ do not contribute very much to the integral.
Since the Newton method (see section 5)
can produce error bounds for the difference
between the true $u_0$ and the computed one and it is well known how
to validate the computation of eigenvectors of compact operators,
it is quite feasible to estimate the errors in the computation and
conclude that indeed the conditions in \equ(explicitnondeg) are
verified for a particular model.
\REMARK If $S$ is an odd trigonometric polynomial and $\epsilon_0$
is a bifurcation point for the hull function, then $-\epsilon_0$ is also a
bifurcation point since $u_{-\epsilon_0} (\theta) = u_{\epsilon_0}(\theta -\pi)$
satisfies the conditions of \clm(branch). Since this symmetry is not
explicitly used in the algorithms, it provides with a useful check of their
accuracy.
\REMARK The compactness of the operator $\calT$ depends crucially on the condition
$\Im \omega \ne 0$. When $\Im \omega = 0$, $\calT$ is not a compact
operator and the bifurcation \clm(bifurcation) does not apply.
To study numerically the domain of analyticity of the map
$\epsilon \mapsto u_\epsilon$, and the nature of the singularities of the
map we will numerically study the domain of analyticity of maps
$\epsilon \to \Gamma [u_\epsilon]$ where $\Gamma$ is an entire map from the
space of continuous functions to the complex numbers.
Clearly the domain of analyticity of
$\epsilon \mapsto \Gamma[ u_\epsilon]$ is not smaller than the
domain of analyticity of the map $\epsilon \mapsto u_\epsilon$.
One expects also that many observables will lead to the same
domain of analyticity, with singularities of the same nature.
Some observables that immediately come
to mind are the evaluation of the function at certain values and
the Fourier coefficients.
\CLAIM{Theorem}(composition)
Let $f : \complex \to C^0$ ($C^0$ the space of continuous functions on
$[-\pi, \pi]$) be a map with $\epsilon = \epsilon_0$ an
isolated branch point of order 2, i.e.
$f(\epsilon) = \sum_{n=0}^{\infty} f_n (\epsilon - \epsilon_0 )^{n/2}, f_1 \ne
0$
where the series converges for $|\epsilon - \epsilon_0| $ small enough.
If $\Gamma : C^0 \to \complex$ is an analytic function in $C^0$,
i.e. for any $g \in C^0$, there exists $\delta = \delta(g) > 0$
such that, whenever $\| h\|_{C^0} < \delta(g)$,
$\Gamma(g + h) = \sum_{k=0}^\infty \frac{1}{k!} D^k \Gamma (g) h^k$
where the series converges uniformly in $h$, and
$D\Gamma (f_0) [f_1] \ne 0$,
then,
the point $\epsilon_0$ is an isolated branch point of order 2 of the
composition map $f \circ \Gamma$.
\PROOF
Since
$f(\epsilon) = \sum_{n=0}^{\infty} f_n (\epsilon - \epsilon_0 )^{n/2}$
converges for $|\epsilon - \epsilon_0| $ small enough,
$\|(\epsilon - \epsilon_0 )^{1/2} \sum_{n=1}^\infty f_n (\epsilon - \epsilon_0)^{(n-1)/2} \|_{C^0}$
can be made arbitrarily small, in particular less than $\delta(f_0)$. Then,
by the analyticity of $\Gamma$,
$$\eqalign{
\Gamma ( f_0 + (\epsilon - \epsilon_0 )^{\frac12} & \sum_{n=1}^\infty f_n
(\epsilon - \epsilon_0)^{\frac{n-1}{2}} ) =
\sum_{k=0}^\infty \frac{1}{k!} D^k \Gamma(f_0)
\left[(\epsilon - \epsilon_0 )^{\frac12} \sum_{n=1}^\infty f_n
(\epsilon - \epsilon_0)^{\frac{n-1}{2}}\right]^k \cr
& = \Gamma(f_0) + D\Gamma (f_0) [(\epsilon - \epsilon_0)^{1/2} f_1] +
(\epsilon - \epsilon_0)
\sum_{n=0}^{\infty} \Gamma_n (\epsilon - \epsilon_0)^{n/2} \cr
& = \Gamma(f_0) +(\epsilon - \epsilon_0)^{1/2} D\Gamma (f_0) [f_1] +
(\epsilon - \epsilon_0)
\sum_{n=0}^{\infty} \Gamma_n (\epsilon - \epsilon_0)^{n/2} \cr}
$$
Since the series for $\Gamma(f)$ converges for $|\epsilon - \epsilon_0|$
small enough, and $D\Gamma (f_0) [ f_1] \ne 0$, $\epsilon_0$
is an isolated branch point of order 2 for the composition map $f \circ \Gamma$
.
\QED
\REMARK If for a map $\Gamma$,
$\Gamma(f)$ is constant for all $\epsilon$, then the composition
map $\epsilon \to f\circ \Gamma$ is actually entire in $\epsilon$,
since $\Gamma(f)$ is independent of $\epsilon$. This is the case
when we take $\Gamma$ to be evaluation at
$\theta = 0, \pi$ for standard-like maps, with $S$ an odd trigonometric
polynomial.
Based on \clm(bifurcation) and our numerical observations
we formulate
the following conjecture about the behavior of the singular points of
the hull function as $\Im \omega \to 0$.
\CLAIM{Conjecture}(conjecture)
If $\Im \omega \ne 0$ the singular points in the analyticity domain of the
hull function for a dense set of standard-like maps
are isolated branch points of order 2.
As $\Im \omega \to 0$,
the branch points move towards the
origin, and accumulate, in the limit of $\omega$ real, diophantine,
to a natural boundary.
We note that a very similar phenomenon of natural boundaries being
approximated by accumulation of branch points was discussed in \cite{LT1}
for a very different problem, namely, invariant curves in a
dissipative system. There, irrational frequencies were approximated by
rational ones and the later were shown to lead to analyticity domains
bounded by branch points. The phenomena in \cite{LT1} were, however,
very different because the derivative of the operator was not
compact, and indeed the spectrum was uncountable.
We also note that the Greene's criterion for complex values \cite{FL}
also suggests that, for twist mappings, the analyticity domain for
invariant curves is approximated by the place at which periodic
orbits lose stability. When the eigenvalues $\lambda_{\pm}$ of the orbit of
type $p/q$ arrive at the unit circle
with a rational phase ($\lambda_{\pm} = e^{2\pi i {N\over M}}$),
one also expects that the periodic orbit of period $ q M$
expressed as a function of the parameter also experiences a branch point
of order two.
\def\T{{\cal R}}
\SECTION Newton Method
In order to verify the perturbative calculations, we will use
a non-perturbative method based on numerical
continuation for an appropriate operator $\T_\epsilon$.
For fixed $\epsilon$, let
$\T_\epsilon: C^{\omega, \delta} \to C^{\omega, \delta}$ with
$$ \T_\epsilon f(\theta) = \epsilon \Delta_\omega^{-1}\bigl [S(f(\theta)+\theta)\bigr ]
-f(\theta ).
\EQ(newtonop)$$
If $u_0$ fails to satisfy $\equ(stdmap)$ by a small amount, i.e.
$$\T_\epsilon u_0(\theta) = R_0(\theta)$$
we can try to improve the approximate solution by setting it to
$u_0(\theta) + \eta (\theta)$,
where $\eta$ will be chosen to make the error much smaller.
\CLAIM{Theorem}(continuation)
Let $\omega$ with $\Im \omega \ne 0, \T_\epsilon$ as in \equ(newtonop),
$\epsilon, u_\epsilon \in C^{\omega, \delta}$ such that
$\T_\epsilon u_\epsilon (\theta) = 0$. If $D\T_\epsilon (u_\epsilon)$ is an
invertible operator with bounded inverse, then , for $|\epsilon - \epsilon '|$
sufficiently small
there exists
$u_{\epsilon '} \in C^{\omega, \delta}$ such that
$\T_{\epsilon '} u_{\epsilon '} (\theta) = 0$.
\PROOF
Since $u_\epsilon$ satisfies $\T_\epsilon u_\epsilon (\theta) = 0$
we have
$$\| \T_{\epsilon '} u_\epsilon (\theta) \|_\delta =
\| \T_{\epsilon '} u_\epsilon (\theta) -\T_{\epsilon} u_\epsilon
(\theta)\|_\delta
\le | \epsilon ' - \epsilon | \| \Delta_\omega^{-1}
S(u_\epsilon(\theta) + \theta)) \|_\delta
$$
Constructing the operator
$$ \Phi (f) = - \left[ D\T_\epsilon (u_\epsilon)\right]^{-1} \T_{\epsilon '}
(f) + f$$
$\Phi$ is a contraction in $\|\ \|_\delta$ of a factor $1/2$ in a
neighborhood of $u_\epsilon$ that can be chosen uniformly for
$|\epsilon ' - \epsilon | $ sufficiently small. Moreover, since
$\| \Phi (u_\epsilon) - u_\epsilon \|_\delta$ can be made as small
as desired by choosing $|\epsilon ' - \epsilon | $ small enough, we
conclude, from the contraction mapping principle, that $\Phi$ has a fixed
point $u_{\epsilon '}$ for all $\epsilon '$ in a neighborhood of $\epsilon$.
But a fixed point of $\Phi$ is a solution of $\T_{\epsilon '} u = 0$.
\QED
\picture{1a}{
The values of the solution of \equ(stdmap) for $\theta = 0.23$ along
a path in the complex plane that encircles a particular point twice.
Set 1 are the values through the first turn, Set 2 the values through the
second turn.
After two turns we come back to the original solution.
Around point $\epsilon = 0.9 + 2.39i$.
$S(q) = \sin( q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$.}
\REMARK
The above theorem does not apply at values of $\epsilon$ where a branch point
can occur, since $D\T_\epsilon (u_\epsilon) $ is not invertible at those
values.
The Newton method has certain advantages over a perturbative method.
A perturbative expansion only converges in a disc of radius bounded by the
position of the singularity closest to the origin,
and does not give any information on the nature of the singularity.
Based on the Newton method, we can perform a continuation method
starting from $\epsilon = 0$, when the
hull function is $u = 0$, along paths in the complex
plane. Initial guesses can be chosen to be either the solutions computed
at points nearby or, for small values of $\epsilon$, the
Lindstedt series \equ(lindstedt).
On the other
hand the Lindstedt series \equ(lindstedt) provides more global
information, that can be used to locate several singular points.
More important for our purposes is that by moving with small steps
around a singularity, we can ascertain its nature.
\picture{1b}{
Same as in figure 1a.
Around point $\epsilon = 0.56 - 2.85i$.
$S(q) = \sin (q) + \sin( 3q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$.}
If the singularity
was indeed a branch point, by going around a closed loop once, we
would move to a different sheet of the Riemann surface. If the
branch point was indeed of order 2, going around the closed loop
twice would bring us back again to the original point.
This prediction has been verified quite unmistakably in figure 1.
Note that the behavior would have been completely different should
the singularity have been a pole, a branch point of some other order
or an essential singularity.
Note also that, by using the continuation method along paths that
wind around the singularities, we could discover the global topology
of the Riemmann surface of the function defined by the Lidstedt
series. We have not done that systematically since we do not have a
clear idea of what we should be expecting and looking for.
Nevertheless there are indications that the Riemann surface becomes
increasingly complicated for large $|\epsilon|$.
\SECTION Pad\'e Approximations
A Pad\'e approximant of order
$[M/N]$ for an
analytic function $f$
is a rational function with numerator $P$ of degree at most $M$
and denominator $Q$
of degree at most $N$ whose Taylor expansion agrees with that
of $f$ up to order $M+N$. We also impose the normalization
condition $Q(0) = 1$.
We refer to \cite{BGM}, \cite{Gi}
for a survey of mathematical results about Pad\'e approximants and
applications in problems of Theoretical Physics.
They have been used in almost all fields in Physics
in which perturbative expansions and their breakdown play a role.
Several authors have recently used Pad\'e approximants for
perturbative expansions of conjugating functions for invariant curves
to estimate domains of analyticity (see \cite{BC}, \cite{BCCF}, \cite{FL},
\cite{BM}, \cite{LT1}).
The coefficients of the numerator and denominator of a
Pad\'e approximant can be computed by
$$f(z) = \frac{P(z)}{Q(z)} + O(z^{N+M+1}),\qquad z\to 0$$
or, equivalently,
$$P(z) = Q(z) f(z) +O(z^{N+M+1}),\qquad z\to 0$$
which results to a linear system of equations for $P_i, Q_i$.
The standard method to examine the domain of analyticity of $f$ is to
compute the poles of the Pad\'e approximants for $f$ and study their behavior.
According to our conjecture the domain of analyticity of $f$
includes branch points of order 2. The presence of branch points
affects the behavior of Pad\'e approximants in ways that are numerically
observable.
In \cite{N1} it was shown that for a certain class of
functions with an even number of branch points of order 2 the poles
and the zeros of the $[N/N]$ Pad\'e approximants accumulate, as $N \to \infty$
, along non-intersecting arcs emanating from the branch points. The position
of the arcs is completely determined by the positions of the branch points.
\picture{2a}{
The poles and zeros of Pad\'e
approximants [25/25] for several values of $\theta$
(Set 1) superimposed with the poles of the Pad\'e
approximant for the derivative of the logarithm of
the derivative of the hull function ($u'$) (Set 2).
$S(q) = \sin(q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.5i$.
}
Numerical investigations (see \cite{HB1,2}) and a conjecture of John
Nutall (see \cite{N2}) suggest that for any function with a finite
number of branch points the zeros and the poles of the
$[N/N]$ Pad\'e approximants accumulate on arcs emanating from the branch
points. Our numerical results support the conjecture (see figures 2, 3).
>From the arguments in section 4, the hull function $u_\epsilon$,
for $\Im \omega \ne 0$, could have
an infinite number of branch points.
Notice that the influence of a singularity that is far from the origin,
at $z = d_2$, on the $n^{\rm th}$ Taylor expansion coefficient decreases
asymptotically like $O(|d_1|^n / |d_2|^n )$, where $d_1$ is the position of
the singularity closest to the origin.
In a finite precision computation only singularities close enough to the origin
can be detected.
To find experimentally
the accuracy necessary to distinguish branch points far from the
origin we have computed Pad\'e
approximants for functions of the form
$$f(z) = \sqrt{1 - \alpha_1 z} + \sqrt{1 + \alpha_1 z} + \sqrt{1 - \alpha_2 z},
\qquad |\alpha_1| > |\alpha_2|.$$
The $[N/N]$ Pad\'e approximant for $f$ gives no indication of a branch point
at $1/\alpha_2$ for $\alpha_2$ small enough (see figure
4 for $\alpha_1 = 1 , \alpha_2 = (5i)^{-1}$) even if we use very high accuracy
in the computation of the coefficients.
\picture{2b}{
Same as figure 2a.
$S(q) = \sin(q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$.
}
\SECTION Improved algorithms based on the nature of the singularities
\leftline{\bf a) Logarithmic Pad\'e approximants}
To locate the position of the singularities we will construct approximants
for functions related to the hull function, with different singular
behavior. The advantage in altering the nature of the singularities
is that certain types of approximants work better for one
kind of singular behavior than another. For example, Pad\'e approximants
can better approximate functions with poles than functions with
branch points.
If a function $f$ has a branch point singularity at $z= 1/\alpha$, then
$$f(z) = A(1-\alpha z)^\gamma + g(z) \EQ(branchfunc)$$
for $g$ analytic at $z=1/\alpha$. If $\gamma < 0$
the contribution from the singularity dominates, and
$f(z) \approx A(1-\alpha z)^\gamma$, for $z$ close to $1/\alpha$.
Notice that the case $\gamma > 0$ (from our
conjecture we expect $\gamma = 1/2$ for
the hull function) reduces to $\tilde \gamma <0$ for
$\tilde f = \frac{d^n f}{dz^n}$, for $\tilde \gamma = \gamma -n,\ n>\gamma$.
Assuming $\gamma < 0$, we form the function
$$F_1 (z) = \frac{d}{dz} \ln f(z) = \frac{f'(z)}{f(z)} \approx
\frac{\gamma}{z - \frac{1}{\alpha}}$$
for $z$ close to $1/\alpha$.
\picture{2c}{
Same as figure 2a.
$S(q) = \sin(q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.05i$.
}
We expect a Pad\'e approximant for $F_1(z)$ to exhibit a pole at
$z = 1/\alpha$ with residue $\gamma$. To form an $[N/M]$ Pad\'e
approximant for $F_1$ we have
$$F_1(z) = f'(z)/f(z) = P(z)/Q(z) + O(z^{N+M+1}), \qquad z \to 0$$
or, provided $f(0)\ne 0, Q(0) = 1$
$$f'(z)Q(z) = f(z)P(z) + O(z^{N+M+1}), \qquad z \to 0.$$
The coefficients of $P, Q$ can be determined by solving a linear system
involving $f_n$ (where $f(z) = \sum_{n=0}^{\infty} f_n z^n$).
The location and the residue of the poles of the Pad\'e approximants for
$F_1$ indicate the location and the order of the branch points of $f$.
Another way to estimate the order of a branch point, once the
location $1/\alpha$ is determined from Pad\'e approximants for $F_1$, is to
form Pad\'e approximants for
$$F_2(z) = (z-\frac{1}{\alpha}) F_1(z) = (z-\frac{1}{\alpha})
\frac{f'(z)}{f(z)} \approx \gamma$$
and
$$F_3(z) =
\frac{\frac{d}{dz} \ln \left[ \frac{d f(z)}{dz} \right]}{\frac{d}{dz} \ln f(z)}
= \frac{f''(z)f(z)}{f'(z)f'(z)} \approx 1 - \frac{1}{\gamma}$$
The accuracy for $\gamma$ depends on the accuracy with which the location
of the branch point is computed. For a discussion of these methods
and applications in which they have been used see \cite{BGM, pg. 55-57},
\cite{HB1,HB2}.
We point out that the presence of the function $g$ in \equ(branchfunc)
can slow the convergence of Pad\'e approximants for $F_1$. The reason is that
for $f$ as in \equ(branchfunc)
$$F_1(z) = \frac{\gamma}{z-\frac{1}{\alpha}} + \gamma \alpha
(1 - \alpha z )^{- \gamma - 1} g(x) + \cdots
\EQ(branchfunc2)$$
\picture{3a}{
Same as figure 2a.
$S(q) = \sin(q )+ \sin(3q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.5i$.
}
The point $z= 1/\alpha$ is not a simple pole for $F_1$. We expect the
Pad\'e approximant for $F_1$ to indicate, apart from a pole, the presence of
a branch point at $1/\alpha$ and the convergence
to $\gamma$ of $[N/N]$ Pad\'e approximants for
$F_2(1/\alpha)$
to be slow.
The second term in \equ(branchfunc2) becomes more important for $|\gamma|$
small.
We can increase the value of $|\gamma |$ if, instead of working
with the function $f$ of \equ(branchfunc), we work with a high order
derivative of $f$.
\leftline{\bf b) Ratio method}
The ratio method is a well known method (see \cite{HB1, HB2}) that
takes advantage of the fact that the coefficients of the
Taylor expansion are strongly influenced by the singularity closest to the
origin. The existence of additional singularities with distances from the
origin close to the radius of convergence greatly reduces the
effectiveness of the method.
\picture{3b}{
Same as figure 2a.
$S(q) = \sin (q) + \sin (3q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$.
}
In the cases we have studied, due to parity properties of $S$, we expect
at least two branch points at the boundary of the domain of convergence
of the Taylor expansion of the hull function at positions
$\epsilon = \pm 1/\alpha$. To study
test functions $f$ with Taylor expansion
$f(z) = \sum_{n=0}^{\infty} f_n z^n$,
$f(z) = (1-\alpha z)^\gamma + (1+\alpha z)^\gamma , \quad \gamma < 0$
we form the ratios
$ r_n = f_{2n}/f_{2(n-1)} $
which converge to $\alpha^2$ with an error of order $1/n$.
We will construct an extrapolation scheme to estimate $\alpha$ more accurately
using the ratios $r_n$.
Consider
$\xi_n = 2n(2n-1) r_n - (2n-2)(2n-3) r_{n-1}$.
Plotting $\xi_n$ versus $n$ gives a straight line with slope
$8\alpha^2$ and intercept $-(4\gamma +10)\alpha^2$. Instead of using
graphical methods to find the slope and the intercept we construct
$$\mu_n = \frac18 (\xi_n - \xi_{n-1} ) = \alpha^2, \quad
\rho_n = -\frac14 (\frac{\xi_n}{\alpha^2} - 8n + 10) = \gamma \EQ(gamma)$$
The value of $\mu_n$ can be substituted for $\alpha^2$ in \equ(gamma),
so that both equations \equ(gamma) depend only on the
coefficients $f_n$.
The sequences $\mu_n$, $\rho_n$ converge to $\alpha$, $\gamma$ and
can be used as independent verification for
the predicted value of $\gamma$.
\picture{3c}{
Same as figure 2a.
$S(q) = \sin (q )+ \sin (3q),\quad \omega = \frac{\sqrt{5}-1}{2} + 0.05i$.
}
If the function
$f$ has additional structure, as we expect for the case of the hull
function, then corrections of order $1/n$ are introduced to \equ(gamma).
Since we expect the formation of a natural boundary in the
limit $\Im \omega \to 0$ the ratio method is not useful for
studying singularities
for $\omega$ real, diophantine, and in general works only for the
singularities closest to the origin. The logarithmic Pad\'e approximants
on the other hand provide information for several branch points simultaneously.
\SECTION Numerical implementation and results
We have used a package we developed (see \cite{LT1})
to manipulate one dimensional Fourier series. The advantage
of the package is the ability to change between double and
extended precision by changing a definitions file. For the extended precision
computations we used the public domain library {\tt PARI/GP}.
In the implementation of the Newton method we truncated the Fourier
series representation for the hull function to mode $n$, for $n$ large
$$u(\theta ) = \sum_{k = -n}^{n} \hat u_k e^{ik\theta}$$
We checked the computation by verifying that it
was converging quadratically. The condition numbers obtained inverting the
derivative matrix, were indicative of the proximity to a branch point.
The paths we chose encircled the points indicated by the Pad\'e methods
several times. Whenever the
method failed to converge quadratically
due to proximity to a branch point, %or due to roundoff error,
we increased the size of the Fourier
series. Notice that if an initial guess is good enough, the Newton method
will converge irrespective of the way the initial guess was chosen.
We computed the coefficients of Pad\'e approximants
using Gaussian elimination and obtained condition
numbers for the computations.
Although recursive methods to compute the Pad\'e approximants
exist (see \cite{BGM}) we do not know of any way to assign condition numbers
to such computations. The actual routines we used were a translation
into {\tt C} of the well known {\tt DECOMP} and {\tt SOLVE} from \cite{FMM}.
To find the zeros of the numerator and denominator of Pad\'e approximants
we used the routines ``{\tt xzroot}'', ``{\tt zroot}'' from \cite{FPTV}
translated to be compatible with the use of extended precision
arithmetic. Although, for Pad\'e approximants in general, the residue of
a pole is indicative of its significance, in the case of functions with branch
points one expects zeros and poles to lie close together, resulting in
large condition numbers
and poles with small residues.
The condition numbers for computing
the coefficients of logarithmic Pad\'e approximants were much smaller
than the ones for straightforward Pad\'e approximants.
\picture{4}{
The poles and the zeros of the [35/35] Pad\'e approximant to
$f(z) = \sqrt{1 - z} + \sqrt{1 + z} + \sqrt{1 - \frac{z}{5i}}$.
Computations performed with 60 digit accuracy.
}
An independent check of the accuracy of our computations is the
symmetry of the branch points. If $\epsilon$ is a branch point, so is
$-\epsilon$ but this symmetry is not built into our numerical method,
thus can be used as an estimate of the accuracy of the computations.
Our numerical results are consistent with the predictions of
section 4. We investigated two standard-like maps,
$S_1(q) = \sin (q), S_2(q) = \sin (q) + \sin (3q)$. Straightforward Pad\'e
approximants for the hull function exhibited poles and zeros
along lines emanating from distinct points.
Logarithmic Pad\'e approximants to the derivative of the hull function
(we used the derivative of the hull function so that,
asymptotically close to the singularities
$u(\epsilon) \approx A(1-\frac{\epsilon}{\epsilon_0} )^{-\frac12})$
exhibited poles at locations consistent with the locations indicated by
the accumulation of poles and zeros of Pad\'e approximants
(see figures 2, 3). The residue
of the poles, computed both from logarithmic Pad\'e methods and
the ratio method, was within $10\%$ of the predicted value
$-1/2$ as indicated in table 1.
The results of the numerical continuation based on the Newton method
support the existence of an isolated branch point of order 2, within
the path followed (see figure 1).
\bigskip
\bigskip
\bigskip
\centerline {\bf Table 1}
\bigskip
\settabs 5 \columns
\+ &\hfill $\alpha$ \hfill &\hfill $\beta$ \hfill &\hfill $\gamma$ \hfill &\hfill $\delta$ \hfill& \cr
\+ \cr
\+ Residue($F_2$) &$-0.53-0.002i$ &$-0.53-0.001i$&$-0.56-0.01i$&$-0.56-0.02i$ \cr
\+ Residue($F_3$) &$-0.52-0.001i$ &$-0.52-0.001i$&$-0.56-0.02i$&$-0.56-0.02i$ \cr
\+ Residue($\rho$)&$-0.500+10^{-7}i$&$-0.500+10^{-6}i$&\hfill --- \hfill&\hfill --- \hfill& \cr
\+ Position($\mu$)&$11.886+14.32i$&$14.47+17.645i$&$0.9 + 2.4i$&$0.6-2.9i$\cr
\bigskip
\bigskip
{\eightpoint \noindent {\bf Table 1:}
Residues and positions of branch points in the domain of analyticity. The
positions of the points $\alpha, \beta, \gamma, \delta$ are computed
from the $F_1$ Pad\'e approximants. The digits reported are accurate
for each method apart from the last digit.
\leftline{$\alpha = 11.889 - 14.320i,\ \omega = \frac{\sqrt{5} - 1}{2} + 0.5i,\ S(q) = \sin(q)$}
\leftline{$\beta = 14.450 + 17.649i,\ \omega = \frac{\sqrt{5} - 1}{2} + 0.5i,\ S(q) = \sin(q)+\sin(3q)$}
\leftline{$\gamma = 0.906 + 2.39i,\ \omega = \frac{\sqrt{5} - 1}{2} + 0.1i,\ S(q) = \sin(q)$}
\leftline{$\delta = 0.56 - 2.85i,\ \omega = \frac{\sqrt{5} - 1}{2} + 0.1i,\ S(q) = \sin(q) +\sin(3q)$}
}
\SECTION Acknowledgments
We would like to thank John Nuttall, Armando Bazzani, and Giorgio
Turchetti for correspondence and discussions.
\SECTION References
\ref
\no{A}
\by{S. Aubry}
\paper{The twist map, the extended Frenkel-Kontorova model and the devil's staircase}
\jour{Physica D}
\vol{7}
\pages{240--258}
\yr{1983}
\endref
\ref
\no{AB}
\by{S. Abenda, A. Bazzani}
\paper{Singularity analysis of 2D complexified hamiltonian systems}
\jour{U. of Bologna preprint}
\endref
\ref
\no{BC}
\by{A. Berretti, L. Chierchia}
\paper{On the complex analytic structure of the golden invariant curve for the standard map}
\jour{Nonlinearity}
\vol{3}\pages{39--44}
\yr{1990}
\endref
\ref
\no{BCCF}
\by{A. Berretti, A. Celletti, L. Chierchia, C. Falcolini}
\paper{Natural boundaries for area preserving twist maps}
\jour{Jour. Stat. Phys.}
\vol{66}
\pages{1613--1630}
\yr{1992}
\endref
\ref
\no{BGM}
\by{G. Baker, M. Graves--Morris}
\book{Pad\'e Approximants}
\publisher{Addison Wesley}
\yr{1981}
\endref
\ref
\no{BM}
\by{A. Berretti, S. Marmi}
\paper{Standard map at complex rotation numbers: Creation of natural boundaries}
\jour{Phys. Rev. Lett.}
\vol{68}
\pages{1443--1446}
\yr{1992}
\endref
\ref
\no{BMT}
\by{L. Billi, M. Malavasi, G. Turchetti}
\paper{Natural boundaries of normalizing transformations}
\jour{U. of Bologna preprint.}
\endref
\ref
\no{BT}
\by{A. Bazzani, G. Turchetti}
\paper{KAM theory of normalizing transformations for complexified area preserving maps}
\jour{U. of Bologna preprint.}
\endref
\ref
\no{CC}
\by{A. Celletti, L. Chierchia}
\paper{Construction of analytic K.A.M. surfaces and effective stability bounds}
\jour{Comm. Math. Phys.}
\vol{118}
\pages{119--161}
\yr{1988}
\endref
\ref
\no{Ch}
\by{B.~V.~Chirikov}
\paper{A universal instability of many-dimensional oscillator systems}
\jour{Phys. Rep.}
\vol{52}
\pages{263--379}
\yr{1979}
\endref
\ref
\no{ChH}
\by{S.~N.~Chow, J.~K.~Hale}
\book{Methods of Bifurcation Theory}
\publisher{Springer--Verlag, New York}
\yr{1982}
\endref
\ref
\no{F}
\by{A. Fathi}
\paper{Une interpr\'etation plus topologique de la d\'emonstration du th\'eoreme du Birkhoff}
\jour{Appendix to \cite{H}.}
\endref
\ref
\no{FL}
\by{C. Falcolini, R. de la Llave}
\paper{Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors}
\jour{Jour. Stat. Phys.}
\vol{67}
\pages{645--666}
\yr{1992}
\endref
\ref
\no{FMM}
\by{G.E. Forsythe, M.A. Malcolm, C. E. Moler}
\book{Computer methods for mathematical computations}
\publisher{Prentice Hall, Englewood Cliffs}
\yr{1977}
\endref
\ref
\no{FPTV}
\by{W.H. Press, B.P. Flannery, S. Teukolski, W. T. Vetterling}
\book{Numerical Recipes}
\publisher{Cambridge Univ. Press, Cambridge}
\yr{1986}
\endref
\ref
\no{Gi}
\by{J. Gilewicz}
\book{Approximants de Pad\'e}
\publisher{Springer--Verlag}
\yr{1978}
\endref
\ref
\no{Gr}
\by{J.~M.~Greene}
\paper{A method for determining a stochastic transition}
\jour{J.~Math.~Phys.}
\vol{20}
\pages{1183--1201}
\yr{1979}
\endref
\ref
\no{GS}
\by{M.~Golubitsky, D.~G.~Schaeffer}
\book{Singularities and groups in bifurcation theory, vol. 1}
\publisher{Springer--Verlag, New York}
\yr{1985}
\endref
\ref
\no{H}
\by{M. R. Herman}
\paper{Sur les courbes invariantes par les diff\'eomorphismes de l'anneau}
\jour{Ast\'erique}
\vol{103--104}
\yr{1983}
\endref
\ref
\no{HB1}
\by{D.L. Hunter, G.A. Baker}
\paper{Methods of Series Analysis I. Comparisons of current methods used in the theory of critical phenomena}
\jour{Phys.~Rev. B}
\vol{7}
\pages{3346--3376}
\yr{1973}
\endref
\ref
\no{HB2}
\by{D.L. Hunter, G.A. Baker}
\paper{Methods of Series Analysis II. Generalized and extended methods with application to the Ising model}
\jour{Phys.~Rev. B}
\vol{7}
\pages{3377--3392}
\yr{1973}
\endref
\ref
\no{LR}
\by{R. de la Llave, D. Rana}
\paper{Accurate strategies for K.~A.~M. bounds and their implementation}
\inbook{Computer Aided proofs in Analysis, K. Meyer, D. Schmidt ed.}
\publisher{Springer Verlag}
\yr{1991}
\endref
\ref
\no{LT1}
\by{R. de la Llave, S. Tompaidis}
\paper{Computation of domains of analyticity for some perturbative expansions from mechanics}
\jour{Physica D}
\vol{71}
\pages{55--81}
\yr{1994}
\endref
\ref
\no{LT2}
\by{R. de la Llave, S. Tompaidis}
\paper{Nature of singularities for analyticity domains of invariant curves}
\yr{Preprint available from {\tt mp\_arc@math.utexas.edu} \# 94-137. Submitted}
\endref
%\ref
% \no{M}
% \by{J. Mather}
% \paper{Non existence of invariant circles}
% \jour{Erg. Th. and Dyn. Syst.}
% \vol{4}
% \pages{301--309}
% \yr{1984}
%\endref
\ref
\no{McK}
\by{R.S. McKay}
\paper{Renormalization in area preserving maps}
\jour{Princeton thesis}
\yr{1982}
\endref
\ref
\no{N1}
\by{J.N. Nuttall}
\paper{The convergence of Pad\'e approximants to functions with branch points}
\inbook{Pad\'e and rational approximation, E.B.~Saff, R.H.~Varga (eds.)}
\pages{101--109}
\publisher{Academic Press, New York}
\yr{1977}
\endref
\ref
\no{N2}
\by{J.N. Nuttall}
\paper{Letter to Stathis Tompaidis, dated January 8, 1993}
\endref
\ref
\no{Po}
\by{H. Poincar\'e}
\book{Les methodes nouvelles de la m\'echanique c\'eleste}
\publisher{Gauthier Villars, Paris}
\yr{1891--1899}
\endref
\ref
\no{R}
\by{D. Rana}
\paper{Proof of accurate upper and lower bounds to stability domains in small denominator problems}
\jour{Princeton thesis}
\yr{1987}
\endref
\ref
\no{Rudin}
\by{W. Rudin}
\book{Functional analysis}
\publisher{McGraw Hill}
\yr{1973}
\endref
\ref
\no{SM}
\by{ C. L. Siegel, J. Moser}
\book{Lectures on Celestial Mechanics}
\publisher{ Springer-Verlag, New York}
\yr{1971}
\endref
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\catcode`\%=12\catcode`\}=12\catcode`\{=12
\catcode`\<=1\catcode`\>=2
\openout6=1a.ps
\write6<%!PS-Adobe-2.0>
\write6<%%Creator: gnuplot>
\write6<%%DocumentFonts: Helvetica>
\write6<%%BoundingBox: 50 50 554 770>
\write6<%%Pages: (atend)>
\write6<%%EndComments>
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6< 0 vshift R show } def>
\write6
\write6< dup stringwidth pop neg vshift R show } def>
\write6
\write6< dup stringwidth pop -2 div vshift R show } def>
\write6
\write6< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6
\write6< currentlinewidth 2 div sub M>
\write6< 0 currentlinewidth V stroke } def>
\write6
\write6< hpt neg vpt neg V hpt vpt neg V>
\write6< hpt vpt V hpt neg vpt V closepath stroke>
\write6< P } def>
\write6
\write6< currentpoint stroke M>
\write6< hpt neg vpt neg R hpt2 0 V stroke>
\write6< } def>
\write6
\write6< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write6< hpt4 neg 0 V closepath stroke>
\write6< P } def>
\write6
\write6< hpt4 vpt4 neg V currentpoint stroke M>
\write6< hpt4 neg 0 R hpt4 vpt4 V stroke } def>
\write6
\write6< hpt neg vpt -1.62 mul V>
\write6< hpt 2 mul 0 V>
\write6< hpt neg vpt 1.62 mul V closepath stroke>
\write6< P } def>
\write6
\write6
\write6<%%EndProlog>
\write6<%%Page: 1 1>
\write6
\write6
\write6<50 50 translate>
\write6<0.100 0.100 scale>
\write6<90 rotate>
\write6<0 -5040 translate>
\write6<0 setgray>
\write6
\write6
\write6
\write6<2883 351 M>
\write6<0 4478 V>
\write6
\write6<840 351 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<756 351 M>
\write6<(-1.8) Rshow>
\write6<840 849 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<756 849 M>
\write6<(-1.6) Rshow>
\write6<840 1346 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-1.4) Rshow>
\write6<840 1844 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-1.2) Rshow>
\write6<840 2341 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-1) Rshow>
\write6<840 2839 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-0.8) Rshow>
\write6<840 3336 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-0.6) Rshow>
\write6<840 3834 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-0.4) Rshow>
\write6<840 4331 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(-0.2) Rshow>
\write6<840 4829 M>
\write6<63 0 V>
\write6<6066 0 R>
\write6<-63 0 V>
\write6<-6150 0 R>
\write6<(0) Rshow>
\write6<840 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<840 211 M>
\write6<(-0.4) Cshow>
\write6<1862 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(-0.2) Cshow>
\write6<2883 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(0) Cshow>
\write6<3905 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(0.2) Cshow>
\write6<4926 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(0.4) Cshow>
\write6<5947 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(0.6) Cshow>
\write6<6969 351 M>
\write6<0 63 V>
\write6<0 4415 R>
\write6<0 -63 V>
\write6<0 -4555 R>
\write6<(0.8) Cshow>
\write6<840 351 M>
\write6<6129 0 V>
\write6<0 4478 V>
\write6<-6129 0 V>
\write6<840 351 L>
\write6<140 2590 M>
\write6
\write6<(Im) Cshow>
\write6
\write6<3904 71 M>
\write6<(Real) Cshow>
\write6<3904 4969 M>
\write6<(Figure 1a) Cshow>
\write6
\write6<6486 4626 M>
\write6<(Set 1) Rshow>
\write6<6654 4626 C>
\write6<2883 4829 C>
\write6<2842 4669 C>
\write6<2803 4509 C>
\write6<2768 4347 C>
\write6<2740 4180 C>
\write6<2722 4003 C>
\write6<2719 3812 C>
\write6<2740 3595 C>
\write6<2800 3337 C>
\write6<2944 2990 C>
\write6<3205 3015 C>
\write6<3461 3024 C>
\write6<3707 3020 C>
\write6<3943 3003 C>
\write6<4169 2976 C>
\write6<4384 2940 C>
\write6<4589 2895 C>
\write6<4786 2842 C>
\write6<4976 2782 C>
\write6<5161 2713 C>
\write6<5344 2635 C>
\write6<5526 2548 C>
\write6<5712 2447 C>
\write6<5903 2328 C>
\write6<6102 2185 C>
\write6<6307 2006 C>
\write6<6511 1766 C>
\write6<6637 1409 C>
\write6<6297 974 C>
\write6<5659 702 C>
\write6
\write6<6486 4486 M>
\write6<(Set 2) Rshow>
\write6<6654 4486 B>
\write6<4991 549 B>
\write6<4336 481 B>
\write6<3716 481 B>
\write6<3152 537 B>
\write6<2656 636 B>
\write6<2234 769 B>
\write6<1886 924 B>
\write6<1610 1094 B>
\write6<1403 1274 B>
\write6<1259 1459 B>
\write6<1176 1644 B>
\write6<1150 1828 B>
\write6<1177 2005 B>
\write6<1252 2174 B>
\write6<1371 2332 B>
\write6<1528 2476 B>
\write6<1718 2606 B>
\write6<1934 2718 B>
\write6<2170 2813 B>
\write6<2422 2890 B>
\write6<2681 2949 B>
\write6<2944 2990 B>
\write6
\write6
\write6
\write6
\write6<%%Trailer>
\write6<%%Pages: 1>
\closeout6
\openout7=1b.ps
\write7<%!PS-Adobe-2.0>
\write7<%%Creator: gnuplot>
\write7<%%DocumentFonts: Helvetica>
\write7<%%BoundingBox: 50 50 554 770>
\write7<%%Pages: (atend)>
\write7<%%EndComments>
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7< 0 vshift R show } def>
\write7
\write7< dup stringwidth pop neg vshift R show } def>
\write7
\write7< dup stringwidth pop -2 div vshift R show } def>
\write7
\write7< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7
\write7< currentlinewidth 2 div sub M>
\write7< 0 currentlinewidth V stroke } def>
\write7
\write7< hpt neg vpt neg V hpt vpt neg V>
\write7< hpt vpt V hpt neg vpt V closepath stroke>
\write7< P } def>
\write7
\write7< currentpoint stroke M>
\write7< hpt neg vpt neg R hpt2 0 V stroke>
\write7< } def>
\write7
\write7< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write7< hpt4 neg 0 V closepath stroke>
\write7< P } def>
\write7
\write7< hpt4 vpt4 neg V currentpoint stroke M>
\write7< hpt4 neg 0 R hpt4 vpt4 V stroke } def>
\write7
\write7< hpt neg vpt -1.62 mul V>
\write7< hpt 2 mul 0 V>
\write7< hpt neg vpt 1.62 mul V closepath stroke>
\write7< P } def>
\write7
\write7
\write7<%%EndProlog>
\write7<%%Page: 1 1>
\write7
\write7
\write7<50 50 translate>
\write7<0.100 0.100 scale>
\write7<90 rotate>
\write7<0 -5040 translate>
\write7<0 setgray>
\write7
\write7
\write7
\write7<840 351 M>
\write7<6129 0 V>
\write7<-1865 0 R>
\write7<0 4478 V>
\write7
\write7<840 351 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<756 351 M>
\write7<(0) Rshow>
\write7<840 849 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<756 849 M>
\write7<(0.1) Rshow>
\write7<840 1346 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.2) Rshow>
\write7<840 1844 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.3) Rshow>
\write7<840 2341 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.4) Rshow>
\write7<840 2839 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.5) Rshow>
\write7<840 3336 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.6) Rshow>
\write7<840 3834 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.7) Rshow>
\write7<840 4331 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.8) Rshow>
\write7<840 4829 M>
\write7<63 0 V>
\write7<6066 0 R>
\write7<-63 0 V>
\write7<-6150 0 R>
\write7<(0.9) Rshow>
\write7<1106 351 M>
\write7<0 63 V>
\write7<0 4415 R>
\write7<0 -63 V>
\write7<0 -4555 R>
\write7<(-0.15) Cshow>
\write7<2439 351 M>
\write7<0 63 V>
\write7<0 4415 R>
\write7<0 -63 V>
\write7<0 -4555 R>
\write7<(-0.1) Cshow>
\write7<3771 351 M>
\write7<0 63 V>
\write7<0 4415 R>
\write7<0 -63 V>
\write7<0 -4555 R>
\write7<(-0.05) Cshow>
\write7<5104 351 M>
\write7<0 63 V>
\write7<0 4415 R>
\write7<0 -63 V>
\write7<0 -4555 R>
\write7<(0) Cshow>
\write7<6436 351 M>
\write7<0 63 V>
\write7<0 4415 R>
\write7<0 -63 V>
\write7<0 -4555 R>
\write7<(0.05) Cshow>
\write7<840 351 M>
\write7<6129 0 V>
\write7<0 4478 V>
\write7<-6129 0 V>
\write7<840 351 L>
\write7<140 2590 M>
\write7
\write7<(Im) Cshow>
\write7
\write7<3904 71 M>
\write7<(Real) Cshow>
\write7<3904 4969 M>
\write7<(Figure 1b) Cshow>
\write7
\write7<6486 4626 M>
\write7<(Set 1) Rshow>
\write7<6654 4626 C>
\write7<5104 351 C>
\write7<5367 820 C>
\write7<5619 1284 C>
\write7<5838 1738 C>
\write7<6008 2179 C>
\write7<6110 2601 C>
\write7<6123 2998 C>
\write7<6025 3363 C>
\write7<5780 3685 C>
\write7<5310 3940 C>
\write7<5624 3991 C>
\write7<5857 4070 C>
\write7<5950 4170 C>
\write7<5861 4281 C>
\write7<5569 4389 C>
\write7<5089 4480 C>
\write7<4461 4542 C>
\write7<3757 4565 C>
\write7<3056 4549 C>
\write7<2436 4498 C>
\write7<1950 4422 C>
\write7<1620 4334 C>
\write7<1434 4245 C>
\write7<1356 4165 C>
\write7<1338 4096 C>
\write7<1337 4037 C>
\write7<1328 3981 C>
\write7<1312 3922 C>
\write7<1315 3854 C>
\write7<1379 3776 C>
\write7
\write7<6486 4486 M>
\write7<(Set 2) Rshow>
\write7<6654 4486 B>
\write7<1542 3693 B>
\write7<1827 3612 B>
\write7<2233 3541 B>
\write7<2739 3490 B>
\write7<3309 3464 B>
\write7<3899 3466 B>
\write7<4464 3495 B>
\write7<4960 3550 B>
\write7<5349 3625 B>
\write7<5602 3713 B>
\write7<5705 3805 B>
\write7<5660 3891 B>
\write7<5489 3963 B>
\write7<5233 4013 B>
\write7<4947 4036 B>
\write7<4688 4035 B>
\write7<4508 4014 B>
\write7<4443 3982 B>
\write7<4509 3948 B>
\write7<4698 3924 B>
\write7<4981 3919 B>
\write7<5310 3940 B>
\write7
\write7
\write7
\write7
\write7<%%Trailer>
\write7<%%Pages: 1>
\closeout7
\openout8=2a.ps
\write8<%!PS-Adobe-2.0>
\write8<%%Creator: gnuplot>
\write8<%%DocumentFonts: Helvetica>
\write8<%%BoundingBox: 50 50 554 770>
\write8<%%Pages: (atend)>
\write8<%%EndComments>
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8< 0 vshift R show } def>
\write8
\write8< dup stringwidth pop neg vshift R show } def>
\write8
\write8< dup stringwidth pop -2 div vshift R show } def>
\write8
\write8< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8
\write8< currentlinewidth 2 div sub M>
\write8< 0 currentlinewidth V stroke } def>
\write8
\write8< hpt neg vpt neg V hpt vpt neg V>
\write8< hpt vpt V hpt neg vpt V closepath stroke>
\write8< P } def>
\write8
\write8< currentpoint stroke M>
\write8< hpt neg vpt neg R hpt2 0 V stroke>
\write8< } def>
\write8
\write8< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write8< hpt4 neg 0 V closepath stroke>
\write8< P } def>
\write8
\write8< hpt2 vpt2 neg V currentpoint stroke M>
\write8< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write8
\write8< hpt neg vpt -1.62 mul V>
\write8< hpt 2 mul 0 V>
\write8< hpt neg vpt 1.62 mul V closepath stroke>
\write8< P } def>
\write8
\write8
\write8<%%EndProlog>
\write8<%%Page: 1 1>
\write8
\write8
\write8<50 50 translate>
\write8<0.100 0.100 scale>
\write8<90 rotate>
\write8<0 -5040 translate>
\write8<0 setgray>
\write8
\write8
\write8
\write8<672 2590 M>
\write8<6297 0 V>
\write8<3821 351 M>
\write8<0 4478 V>
\write8
\write8<672 351 M>
\write8<63 0 V>
\write8<6234 0 R>
\write8<-63 0 V>
\write8<588 351 M>
\write8<(-100) Rshow>
\write8<672 1471 M>
\write8<63 0 V>
\write8<6234 0 R>
\write8<-63 0 V>
\write8<-6318 0 R>
\write8<(-50) Rshow>
\write8<672 2590 M>
\write8<63 0 V>
\write8<6234 0 R>
\write8<-63 0 V>
\write8<-6318 0 R>
\write8<(0) Rshow>
\write8<672 3710 M>
\write8<63 0 V>
\write8<6234 0 R>
\write8<-63 0 V>
\write8<-6318 0 R>
\write8<(50) Rshow>
\write8<672 4829 M>
\write8<63 0 V>
\write8<6234 0 R>
\write8<-63 0 V>
\write8<-6318 0 R>
\write8<(100) Rshow>
\write8<672 351 M>
\write8<0 63 V>
\write8<0 4415 R>
\write8<0 -63 V>
\write8<672 211 M>
\write8<(-100) Cshow>
\write8<2246 351 M>
\write8<0 63 V>
\write8<0 4415 R>
\write8<0 -63 V>
\write8<0 -4555 R>
\write8<(-50) Cshow>
\write8<3821 351 M>
\write8<0 63 V>
\write8<0 4415 R>
\write8<0 -63 V>
\write8<0 -4555 R>
\write8<(0) Cshow>
\write8<5395 351 M>
\write8<0 63 V>
\write8<0 4415 R>
\write8<0 -63 V>
\write8<0 -4555 R>
\write8<(50) Cshow>
\write8<6969 351 M>
\write8<0 63 V>
\write8<0 4415 R>
\write8<0 -63 V>
\write8<0 -4555 R>
\write8<(100) Cshow>
\write8<672 351 M>
\write8<6297 0 V>
\write8<0 4478 V>
\write8<-6297 0 V>
\write8<672 351 L>
\write8<3820 71 M>
\write8<(Im epsilon) Cshow>
\write8<3820 4969 M>
\write8<(Figure 2a) Cshow>
\write8
\write8<6486 4626 M>
\write8<(Set 1) Rshow>
\write8<6654 4626 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<4215 2929 C>
\write8<3426 2252 C>
\write8<4218 2929 C>
\write8<3421 2247 C>
\write8<3398 2228 C>
\write8<4246 2956 C>
\write8<4283 2988 C>
\write8<3355 2191 C>
\write8<4332 3031 C>
\write8<3309 2150 C>
\write8<4405 3095 C>
\write8<3234 2085 C>
\write8<4515 3192 C>
\write8<3123 1988 C>
\write8<4694 3349 C>
\write8<2942 1831 C>
\write8<5022 3636 C>
\write8<2607 1542 C>
\write8<5792 4307 C>
\write8<1802 856 C>
\write8<3821 2590 C>
\write8<3446 2269 C>
\write8<4196 2912 C>
\write8<4214 2927 C>
\write8<3424 2250 C>
\write8<3423 2250 C>
\write8<4220 2933 C>
\write8<3389 2220 C>
\write8<4252 2961 C>
\write8<4290 2994 C>
\write8<3350 2186 C>
\write8<4345 3043 C>
\write8<3294 2138 C>
\write8<4428 3116 C>
\write8<3211 2065 C>
\write8<4557 3229 C>
\write8<3079 1951 C>
\write8<4777 3425 C>
\write8<2854 1755 C>
\write8<5224 3821 C>
\write8<2390 1357 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<3433 2246 C>
\write8<4215 2929 C>
\write8<4217 2929 C>
\write8<3417 2256 C>
\write8<4246 2956 C>
\write8<3393 2223 C>
\write8<3358 2192 C>
\write8<4284 2988 C>
\write8<4332 3031 C>
\write8<3308 2149 C>
\write8<4406 3096 C>
\write8<3235 2085 C>
\write8<4517 3192 C>
\write8<3124 1988 C>
\write8<4697 3349 C>
\write8<2943 1831 C>
\write8<5029 3638 C>
\write8<2609 1543 C>
\write8<5821 4314 C>
\write8<1807 861 C>
\write8<3821 2590 C>
\write8<3446 2269 C>
\write8<4196 2912 C>
\write8<4214 2927 C>
\write8<3424 2249 C>
\write8<3423 2250 C>
\write8<4220 2933 C>
\write8<3390 2220 C>
\write8<4252 2961 C>
\write8<4290 2994 C>
\write8<3350 2186 C>
\write8<4346 3043 C>
\write8<3295 2137 C>
\write8<4429 3116 C>
\write8<3211 2064 C>
\write8<4559 3230 C>
\write8<3081 1950 C>
\write8<4781 3426 C>
\write8<2856 1755 C>
\write8<5237 3824 C>
\write8<2395 1358 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<3433 2246 C>
\write8<4216 2929 C>
\write8<4217 2929 C>
\write8<3416 2256 C>
\write8<4247 2956 C>
\write8<3393 2222 C>
\write8<3357 2192 C>
\write8<4285 2988 C>
\write8<4334 3031 C>
\write8<3307 2149 C>
\write8<4409 3096 C>
\write8<3232 2084 C>
\write8<4521 3192 C>
\write8<3120 1987 C>
\write8<4703 3350 C>
\write8<2937 1829 C>
\write8<5040 3638 C>
\write8<2600 1539 C>
\write8<5846 4319 C>
\write8<1792 856 C>
\write8<3821 2590 C>
\write8<4195 2911 C>
\write8<3445 2268 C>
\write8<3427 2253 C>
\write8<4217 2930 C>
\write8<4218 2930 C>
\write8<3421 2247 C>
\write8<4252 2960 C>
\write8<3388 2219 C>
\write8<3350 2186 C>
\write8<4292 2994 C>
\write8<4348 3043 C>
\write8<3293 2137 C>
\write8<4432 3116 C>
\write8<3209 2064 C>
\write8<4564 3230 C>
\write8<3077 1949 C>
\write8<4790 3426 C>
\write8<2850 1753 C>
\write8<5254 3824 C>
\write8<2386 1353 C>
\write8<4194 2910 C>
\write8<3446 2269 C>
\write8<4208 2934 C>
\write8<3426 2251 C>
\write8<3423 2251 C>
\write8<4224 2924 C>
\write8<3395 2224 C>
\write8<4248 2957 C>
\write8<4283 2988 C>
\write8<3357 2192 C>
\write8<3309 2149 C>
\write8<4332 3031 C>
\write8<3236 2085 C>
\write8<4406 3095 C>
\write8<3126 1989 C>
\write8<4516 3192 C>
\write8<2946 1833 C>
\write8<4696 3349 C>
\write8<2616 1546 C>
\write8<5029 3637 C>
\write8<1831 879 C>
\write8<5829 4322 C>
\write8<3821 2590 C>
\write8<4195 2911 C>
\write8<3445 2268 C>
\write8<3427 2253 C>
\write8<4217 2930 C>
\write8<4217 2931 C>
\write8<3421 2247 C>
\write8<4251 2960 C>
\write8<3389 2219 C>
\write8<3352 2186 C>
\write8<4291 2994 C>
\write8<3296 2137 C>
\write8<4346 3043 C>
\write8<3213 2065 C>
\write8<4429 3116 C>
\write8<3084 1951 C>
\write8<4559 3229 C>
\write8<2863 1757 C>
\write8<4783 3424 C>
\write8<2410 1363 C>
\write8<5243 3822 C>
\write8<4194 2911 C>
\write8<3446 2268 C>
\write8<4206 2935 C>
\write8<3428 2248 C>
\write8<3423 2253 C>
\write8<4224 2923 C>
\write8<3397 2219 C>
\write8<4246 2962 C>
\write8<4279 2990 C>
\write8<3361 2188 C>
\write8<4324 3037 C>
\write8<3317 2141 C>
\write8<3250 2075 C>
\write8<4391 3103 C>
\write8<3151 1976 C>
\write8<4493 3201 C>
\write8<2987 1818 C>
\write8<4661 3358 C>
\write8<2681 1528 C>
\write8<4975 3647 C>
\write8<1957 851 C>
\write8<5736 4333 C>
\write8<3821 2590 C>
\write8<4195 2912 C>
\write8<3445 2268 C>
\write8<3427 2254 C>
\write8<4215 2933 C>
\write8<3423 2244 C>
\write8<4219 2928 C>
\write8<4248 2965 C>
\write8<3391 2216 C>
\write8<3356 2181 C>
\write8<4286 2998 C>
\write8<3305 2129 C>
\write8<4336 3049 C>
\write8<3231 2053 C>
\write8<4412 3124 C>
\write8<3114 1937 C>
\write8<4532 3239 C>
\write8<2912 1741 C>
\write8<4742 3434 C>
\write8<2496 1341 C>
\write8<5177 3832 C>
\write8<4195 2910 C>
\write8<3446 2269 C>
\write8<4208 2934 C>
\write8<3425 2250 C>
\write8<3424 2252 C>
\write8<4225 2924 C>
\write8<3393 2225 C>
\write8<4250 2956 C>
\write8<4285 2987 C>
\write8<3356 2193 C>
\write8<3305 2151 C>
\write8<4336 3029 C>
\write8<3230 2088 C>
\write8<4412 3093 C>
\write8<3117 1992 C>
\write8<4525 3188 C>
\write8<2934 1838 C>
\write8<4710 3343 C>
\write8<2596 1554 C>
\write8<5051 3629 C>
\write8<1805 892 C>
\write8<5869 4308 C>
\write8<3821 2590 C>
\write8<4196 2911 C>
\write8<3445 2268 C>
\write8<3428 2253 C>
\write8<4216 2930 C>
\write8<4219 2930 C>
\write8<3420 2248 C>
\write8<3387 2220 C>
\write8<4254 2959 C>
\write8<3349 2188 C>
\write8<4293 2993 C>
\write8<3292 2139 C>
\write8<4350 3041 C>
\write8<3207 2068 C>
\write8<4436 3113 C>
\write8<3074 1955 C>
\write8<4570 3225 C>
\write8<2848 1763 C>
\write8<4801 3418 C>
\write8<2387 1371 C>
\write8<5273 3812 C>
\write8<4194 2910 C>
\write8<3446 2269 C>
\write8<4208 2934 C>
\write8<3425 2251 C>
\write8<3424 2251 C>
\write8<4224 2924 C>
\write8<3394 2224 C>
\write8<4250 2957 C>
\write8<4284 2988 C>
\write8<3356 2192 C>
\write8<3306 2149 C>
\write8<4335 3031 C>
\write8<3231 2085 C>
\write8<4410 3095 C>
\write8<3118 1988 C>
\write8<4523 3192 C>
\write8<2935 1831 C>
\write8<4707 3349 C>
\write8<2597 1542 C>
\write8<5047 3638 C>
\write8<1794 866 C>
\write8<5861 4320 C>
\write8<3821 2590 C>
\write8<4196 2911 C>
\write8<3445 2268 C>
\write8<3428 2253 C>
\write8<4217 2930 C>
\write8<4218 2931 C>
\write8<3420 2247 C>
\write8<4253 2960 C>
\write8<3388 2219 C>
\write8<3349 2186 C>
\write8<4292 2994 C>
\write8<3292 2137 C>
\write8<4349 3043 C>
\write8<3208 2064 C>
\write8<4434 3116 C>
\write8<3075 1950 C>
\write8<4567 3230 C>
\write8<2848 1755 C>
\write8<4796 3425 C>
\write8<2385 1357 C>
\write8<5265 3823 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<3434 2246 C>
\write8<4215 2929 C>
\write8<4217 2929 C>
\write8<3417 2256 C>
\write8<4246 2955 C>
\write8<3393 2223 C>
\write8<3358 2193 C>
\write8<4284 2988 C>
\write8<4333 3030 C>
\write8<3308 2150 C>
\write8<4406 3094 C>
\write8<3235 2086 C>
\write8<4517 3190 C>
\write8<3124 1990 C>
\write8<4698 3346 C>
\write8<2943 1834 C>
\write8<5032 3633 C>
\write8<2609 1548 C>
\write8<5831 4309 C>
\write8<1808 871 C>
\write8<3821 2590 C>
\write8<3446 2269 C>
\write8<4196 2912 C>
\write8<4214 2927 C>
\write8<3424 2250 C>
\write8<3423 2250 C>
\write8<4220 2933 C>
\write8<3390 2220 C>
\write8<4252 2960 C>
\write8<4290 2993 C>
\write8<3350 2186 C>
\write8<3295 2138 C>
\write8<4346 3042 C>
\write8<4429 3114 C>
\write8<3211 2066 C>
\write8<4560 3228 C>
\write8<3081 1953 C>
\write8<4784 3422 C>
\write8<2856 1759 C>
\write8<5243 3817 C>
\write8<2397 1364 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<3434 2246 C>
\write8<4215 2929 C>
\write8<4217 2929 C>
\write8<3417 2256 C>
\write8<4246 2956 C>
\write8<3393 2223 C>
\write8<3358 2193 C>
\write8<4284 2988 C>
\write8<4333 3031 C>
\write8<3308 2149 C>
\write8<4407 3095 C>
\write8<3234 2085 C>
\write8<4518 3191 C>
\write8<3123 1989 C>
\write8<4699 3348 C>
\write8<2942 1832 C>
\write8<5032 3636 C>
\write8<2607 1544 C>
\write8<5825 4309 C>
\write8<1803 863 C>
\write8<3821 2590 C>
\write8<3446 2269 C>
\write8<4196 2912 C>
\write8<4214 2927 C>
\write8<3424 2250 C>
\write8<3423 2250 C>
\write8<4220 2933 C>
\write8<3390 2220 C>
\write8<4252 2961 C>
\write8<4290 2993 C>
\write8<3350 2186 C>
\write8<4346 3043 C>
\write8<3295 2138 C>
\write8<4430 3115 C>
\write8<3211 2065 C>
\write8<4560 3229 C>
\write8<3079 1952 C>
\write8<4783 3424 C>
\write8<2854 1757 C>
\write8<5240 3821 C>
\write8<2391 1360 C>
\write8<3447 2270 C>
\write8<4195 2911 C>
\write8<3434 2246 C>
\write8<4215 2929 C>
\write8<4218 2929 C>
\write8<3417 2256 C>
\write8<4246 2957 C>
\write8<3393 2222 C>
\write8<3358 2192 C>
\write8<4284 2989 C>
\write8<4332 3032 C>
\write8<3308 2148 C>
\write8<4405 3097 C>
\write8<3234 2084 C>
\write8<4514 3195 C>
\write8<3123 1987 C>
\write8<4691 3353 C>
\write8<2942 1830 C>
\write8<5015 3644 C>
\write8<2607 1540 C>
\write8<5771 4323 C>
\write8<1799 855 C>
\write8<3821 2590 C>
\write8<3446 2269 C>
\write8<4196 2912 C>
\write8<4214 2926 C>
\write8<3425 2250 C>
\write8<3423 2249 C>
\write8<4220 2933 C>
\write8<3389 2219 C>
\write8<4252 2961 C>
\write8<4289 2995 C>
\write8<3350 2185 C>
\write8<4345 3044 C>
\write8<3294 2137 C>
\write8<4427 3118 C>
\write8<3211 2063 C>
\write8<4555 3233 C>
\write8<3080 1949 C>
\write8<4772 3430 C>
\write8<2854 1754 C>
\write8<5213 3831 C>
\write8<2389 1355 C>
\write8
\write8<6486 4486 M>
\write8<(Set 2) Rshow>
\write8<6654 4486 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8<3446 2269 B>
\write8<3446 2269 B>
\write8<4195 2911 B>
\write8
\write8
\write8
\write8
\write8<%%Trailer>
\write8<%%Pages: 1>
\closeout8
\openout9=2b.ps
\write9<%!PS-Adobe-2.0>
\write9<%%Creator: gnuplot>
\write9<%%DocumentFonts: Helvetica>
\write9<%%BoundingBox: 50 50 554 770>
\write9<%%Pages: (atend)>
\write9<%%EndComments>
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9< 0 vshift R show } def>
\write9
\write9< dup stringwidth pop neg vshift R show } def>
\write9
\write9< dup stringwidth pop -2 div vshift R show } def>
\write9
\write9< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9
\write9< currentlinewidth 2 div sub M>
\write9< 0 currentlinewidth V stroke } def>
\write9
\write9< hpt neg vpt neg V hpt vpt neg V>
\write9< hpt vpt V hpt neg vpt V closepath stroke>
\write9< P } def>
\write9
\write9< currentpoint stroke M>
\write9< hpt neg vpt neg R hpt2 0 V stroke>
\write9< } def>
\write9
\write9< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write9< hpt4 neg 0 V closepath stroke>
\write9< P } def>
\write9
\write9< hpt2 vpt2 neg V currentpoint stroke M>
\write9< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write9
\write9< hpt neg vpt -1.62 mul V>
\write9< hpt 2 mul 0 V>
\write9< hpt neg vpt 1.62 mul V closepath stroke>
\write9< P } def>
\write9
\write9
\write9<%%EndProlog>
\write9<%%Page: 1 1>
\write9
\write9
\write9<50 50 translate>
\write9<0.100 0.100 scale>
\write9<90 rotate>
\write9<0 -5040 translate>
\write9<0 setgray>
\write9
\write9
\write9
\write9<672 2590 M>
\write9<6297 0 V>
\write9<3821 351 M>
\write9<0 4478 V>
\write9
\write9<672 351 M>
\write9<63 0 V>
\write9<6234 0 R>
\write9<-63 0 V>
\write9<588 351 M>
\write9<(-10) Rshow>
\write9<672 1471 M>
\write9<63 0 V>
\write9<6234 0 R>
\write9<-63 0 V>
\write9<-6318 0 R>
\write9<(-5) Rshow>
\write9<672 2590 M>
\write9<63 0 V>
\write9<6234 0 R>
\write9<-63 0 V>
\write9<-6318 0 R>
\write9<(0) Rshow>
\write9<672 3710 M>
\write9<63 0 V>
\write9<6234 0 R>
\write9<-63 0 V>
\write9<-6318 0 R>
\write9<(5) Rshow>
\write9<672 4829 M>
\write9<63 0 V>
\write9<6234 0 R>
\write9<-63 0 V>
\write9<-6318 0 R>
\write9<(10) Rshow>
\write9<672 351 M>
\write9<0 63 V>
\write9<0 4415 R>
\write9<0 -63 V>
\write9<672 211 M>
\write9<(-10) Cshow>
\write9<2246 351 M>
\write9<0 63 V>
\write9<0 4415 R>
\write9<0 -63 V>
\write9<0 -4555 R>
\write9<(-5) Cshow>
\write9<3821 351 M>
\write9<0 63 V>
\write9<0 4415 R>
\write9<0 -63 V>
\write9<0 -4555 R>
\write9<(0) Cshow>
\write9<5395 351 M>
\write9<0 63 V>
\write9<0 4415 R>
\write9<0 -63 V>
\write9<0 -4555 R>
\write9<(5) Cshow>
\write9<6969 351 M>
\write9<0 63 V>
\write9<0 4415 R>
\write9<0 -63 V>
\write9<0 -4555 R>
\write9<(10) Cshow>
\write9<672 351 M>
\write9<6297 0 V>
\write9<0 4478 V>
\write9<-6297 0 V>
\write9<672 351 L>
\write9<3820 71 M>
\write9<(Im epsilon) Cshow>
\write9<3820 4969 M>
\write9<(Figure 2b) Cshow>
\write9
\write9<6486 4626 M>
\write9<(Set 1) Rshow>
\write9<6654 4626 C>
\write9<4038 3378 C>
\write9<4107 3131 C>
\write9<3531 2049 C>
\write9<4112 3154 C>
\write9<4115 3271 C>
\write9<4117 3197 C>
\write9<4120 3798 C>
\write9<3518 2028 C>
\write9<3493 1989 C>
\write9<3446 1923 C>
\write9<3356 1814 C>
\write9<3153 1615 C>
\write9<4666 2117 C>
\write9<2970 3065 C>
\write9<2931 2852 C>
\write9<4724 2080 C>
\write9<2894 3112 C>
\write9<4852 1999 C>
\write9<2712 3236 C>
\write9<5166 1806 C>
\write9<2058 3781 C>
\write9<5687 2985 C>
\write9<1852 2154 C>
\write9<3821 2590 C>
\write9<4040 3195 C>
\write9<4048 3368 C>
\write9<4100 3236 C>
\write9<4105 3163 C>
\write9<4106 3133 C>
\write9<3531 2049 C>
\write9<3516 2025 C>
\write9<3485 1980 C>
\write9<4173 4188 C>
\write9<3425 1904 C>
\write9<3293 1774 C>
\write9<4374 3404 C>
\write9<4669 2116 C>
\write9<2967 3067 C>
\write9<2931 2852 C>
\write9<4740 2075 C>
\write9<2891 1590 C>
\write9<2876 3128 C>
\write9<4909 1984 C>
\write9<2631 3314 C>
\write9<5460 1762 C>
\write9<5825 3036 C>
\write9<1738 1927 C>
\write9<3576 1560 C>
\write9<3553 1122 C>
\write9<3533 2049 C>
\write9<4109 3131 C>
\write9<3525 2027 C>
\write9<4119 3152 C>
\write9<3512 1985 C>
\write9<4137 3193 C>
\write9<3494 1912 C>
\write9<3481 1780 C>
\write9<4170 3263 C>
\write9<4231 3393 C>
\write9<4409 3702 C>
\write9<4662 2114 C>
\write9<2975 3063 C>
\write9<4704 2067 C>
\write9<2920 3104 C>
\write9<4772 1968 C>
\write9<2799 3194 C>
\write9<4919 1754 C>
\write9<2498 3423 C>
\write9<2019 2201 C>
\write9<5672 3076 C>
\write9<3821 2590 C>
\write9<3627 1485 C>
\write9<4108 3132 C>
\write9<3533 2048 C>
\write9<4117 3156 C>
\write9<3524 2024 C>
\write9<3510 1976 C>
\write9<4131 3203 C>
\write9<3493 1726 C>
\write9<3491 1891 C>
\write9<4151 3286 C>
\write9<4173 3439 C>
\write9<4202 3829 C>
\write9<4664 2114 C>
\write9<2975 3066 C>
\write9<4712 2064 C>
\write9<2916 3116 C>
\write9<4794 1960 C>
\write9<2792 3244 C>
\write9<2648 3670 C>
\write9<4999 1744 C>
\write9<2010 2194 C>
\write9<5638 3087 C>
\write9<4109 3131 C>
\write9<3532 2049 C>
\write9<4119 3152 C>
\write9<3521 2028 C>
\write9<4139 3192 C>
\write9<3500 1988 C>
\write9<4173 3260 C>
\write9<3462 1920 C>
\write9<4236 3379 C>
\write9<3386 1800 C>
\write9<4397 3621 C>
\write9<3177 1572 C>
\write9<4664 2116 C>
\write9<2977 3066 C>
\write9<4714 2074 C>
\write9<2927 3118 C>
\write9<2826 3245 C>
\write9<4816 1979 C>
\write9<5027 1744 C>
\write9<2612 3617 C>
\write9<5354 4407 C>
\write9<2092 1034 C>
\write9<2012 2141 C>
\write9<5648 3025 C>
\write9<3821 2590 C>
\write9<4109 3131 C>
\write9<3532 2049 C>
\write9<4120 3155 C>
\write9<3521 2025 C>
\write9<4141 3200 C>
\write9<3500 1980 C>
\write9<4177 3278 C>
\write9<3461 1900 C>
\write9<4249 3426 C>
\write9<3381 1751 C>
\write9<4478 3777 C>
\write9<3105 1431 C>
\write9<4663 2114 C>
\write9<2977 3067 C>
\write9<4708 2065 C>
\write9<2931 3120 C>
\write9<4780 1957 C>
\write9<2843 3242 C>
\write9<4913 1748 C>
\write9<2623 3564 C>
\write9<1963 2135 C>
\write9<5696 3024 C>
\write9<4109 3131 C>
\write9<3531 2049 C>
\write9<4119 3152 C>
\write9<3518 2028 C>
\write9<4138 3193 C>
\write9<3490 1987 C>
\write9<4169 3263 C>
\write9<3420 1900 C>
\write9<4226 3388 C>
\write9<3368 1945 C>
\write9<3364 1699 C>
\write9<4352 3672 C>
\write9<4662 2117 C>
\write9<2977 3058 C>
\write9<4706 2081 C>
\write9<2930 3084 C>
\write9<4799 2006 C>
\write9<2833 3150 C>
\write9<5019 1839 C>
\write9<2615 3320 C>
\write9<1924 2165 C>
\write9<5729 3022 C>
\write9<6142 1127 C>
\write9<3821 2590 C>
\write9<4109 3132 C>
\write9<3532 2048 C>
\write9<3525 1873 C>
\write9<4121 3156 C>
\write9<3520 2022 C>
\write9<4142 3202 C>
\write9<3498 1966 C>
\write9<4181 3284 C>
\write9<3448 1746 C>
\write9<4260 3445 C>
\write9<3364 1945 C>
\write9<4610 3907 C>
\write9<4663 2115 C>
\write9<2975 3058 C>
\write9<4710 2071 C>
\write9<2921 3086 C>
\write9<4815 1974 C>
\write9<2802 3156 C>
\write9<5088 1690 C>
\write9<2487 3348 C>
\write9<2091 727 C>
\write9<1929 2017 C>
\write9<5738 3097 C>
\write9<3609 1690 C>
\write9<3564 1318 C>
\write9<3534 2049 C>
\write9<4110 3131 C>
\write9<3528 2026 C>
\write9<3521 1983 C>
\write9<3518 1908 C>
\write9<4124 3153 C>
\write9<4152 3194 C>
\write9<4206 3264 C>
\write9<4331 3380 C>
\write9<4613 3485 C>
\write9<4664 2545 C>
\write9<2975 3063 C>
\write9<4669 2115 C>
\write9<2916 3101 C>
\write9<4738 2069 C>
\write9<2786 3185 C>
\write9<4898 1951 C>
\write9<2456 3384 C>
\write9<5379 1454 C>
\write9<1958 2201 C>
\write9<5807 3051 C>
\write9<3821 2590 C>
\write9<3872 836 C>
\write9<3611 1667 C>
\write9<3606 2012 C>
\write9<3534 2046 C>
\write9<3532 2012 C>
\write9<4111 3132 C>
\write9<3517 1937 C>
\write9<4127 3156 C>
\write9<4161 3203 C>
\write9<4234 3285 C>
\write9<3328 1895 C>
\write9<4432 3404 C>
\write9<4664 2545 C>
\write9<2971 3064 C>
\write9<4672 2113 C>
\write9<2899 3107 C>
\write9<4753 2054 C>
\write9<4755 3465 C>
\write9<2723 3204 C>
\write9<4967 1881 C>
\write9<2092 3432 C>
\write9<1823 2171 C>
\write9<5890 3382 C>
\write9<3603 1802 C>
\write9<3534 2049 C>
\write9<4110 3131 C>
\write9<3529 2026 C>
\write9<3526 1909 C>
\write9<3524 1983 C>
\write9<3521 1382 C>
\write9<4123 3152 C>
\write9<4148 3191 C>
\write9<4195 3257 C>
\write9<4285 3366 C>
\write9<4488 3565 C>
\write9<2975 3063 C>
\write9<4671 2115 C>
\write9<4710 2328 C>
\write9<2917 3100 C>
\write9<4747 2068 C>
\write9<2789 3181 C>
\write9<4929 1944 C>
\write9<2475 3374 C>
\write9<5583 1399 C>
\write9<1954 2195 C>
\write9<5789 3026 C>
\write9<3821 2590 C>
\write9<3601 1985 C>
\write9<3593 1812 C>
\write9<3541 1944 C>
\write9<3536 2017 C>
\write9<3535 2047 C>
\write9<4110 3131 C>
\write9<4125 3155 C>
\write9<4156 3200 C>
\write9<3468 992 C>
\write9<4216 3276 C>
\write9<4348 3406 C>
\write9<3267 1776 C>
\write9<2972 3064 C>
\write9<4674 2113 C>
\write9<4710 2328 C>
\write9<2901 3105 C>
\write9<4750 3590 C>
\write9<4765 2052 C>
\write9<2732 3196 C>
\write9<5010 1866 C>
\write9<2181 3418 C>
\write9<1816 2144 C>
\write9<5903 3253 C>
\write9<4065 3620 C>
\write9<4088 4058 C>
\write9<4108 3131 C>
\write9<3532 2049 C>
\write9<4116 3153 C>
\write9<3522 2028 C>
\write9<4129 3195 C>
\write9<3504 1987 C>
\write9<4147 3268 C>
\write9<4160 3400 C>
\write9<3471 1917 C>
\write9<3410 1787 C>
\write9<3232 1478 C>
\write9<2979 3066 C>
\write9<4666 2117 C>
\write9<2937 3113 C>
\write9<4721 2076 C>
\write9<2869 3212 C>
\write9<4842 1986 C>
\write9<2722 3426 C>
\write9<5143 1757 C>
\write9<5622 2979 C>
\write9<1969 2104 C>
\write9<3821 2590 C>
\write9<4014 3695 C>
\write9<3533 2048 C>
\write9<4108 3132 C>
\write9<3524 2024 C>
\write9<4117 3156 C>
\write9<4131 3204 C>
\write9<3510 1977 C>
\write9<4148 3454 C>
\write9<4150 3289 C>
\write9<3490 1894 C>
\write9<3468 1741 C>
\write9<3439 1351 C>
\write9<2977 3066 C>
\write9<4666 2114 C>
\write9<2929 3116 C>
\write9<4725 2064 C>
\write9<2847 3220 C>
\write9<4849 1936 C>
\write9<4993 1510 C>
\write9<2642 3436 C>
\write9<5631 2986 C>
\write9<2003 2093 C>
\write9<3532 2049 C>
\write9<4109 3131 C>
\write9<3522 2028 C>
\write9<4120 3152 C>
\write9<3502 1988 C>
\write9<4141 3192 C>
\write9<3468 1920 C>
\write9<4179 3260 C>
\write9<3405 1801 C>
\write9<4255 3380 C>
\write9<3244 1559 C>
\write9<4464 3608 C>
\write9<2977 3064 C>
\write9<4664 2114 C>
\write9<2927 3106 C>
\write9<4714 2062 C>
\write9<4815 1935 C>
\write9<2825 3201 C>
\write9<2614 3436 C>
\write9<5029 1563 C>
\write9<2287 773 C>
\write9<5549 4146 C>
\write9<5629 3039 C>
\write9<1993 2155 C>
\write9<3821 2590 C>
\write9<3532 2049 C>
\write9<4109 3131 C>
\write9<3521 2025 C>
\write9<4120 3155 C>
\write9<3500 1980 C>
\write9<4141 3200 C>
\write9<3464 1902 C>
\write9<4180 3280 C>
\write9<3392 1754 C>
\write9<4260 3429 C>
\write9<3163 1403 C>
\write9<4536 3749 C>
\write9<2978 3066 C>
\write9<4664 2113 C>
\write9<2933 3115 C>
\write9<4710 2060 C>
\write9<2861 3223 C>
\write9<4798 1938 C>
\write9<2728 3432 C>
\write9<5018 1616 C>
\write9<5678 3045 C>
\write9<1945 2156 C>
\write9<3532 2049 C>
\write9<4110 3131 C>
\write9<3522 2028 C>
\write9<4123 3152 C>
\write9<3503 1987 C>
\write9<4151 3193 C>
\write9<3472 1917 C>
\write9<4221 3280 C>
\write9<3415 1792 C>
\write9<4273 3235 C>
\write9<4277 3481 C>
\write9<3289 1508 C>
\write9<2979 3063 C>
\write9<4664 2122 C>
\write9<2935 3099 C>
\write9<4711 2096 C>
\write9<2842 3174 C>
\write9<4808 2030 C>
\write9<2622 3341 C>
\write9<5026 1860 C>
\write9<5717 3015 C>
\write9<1912 2158 C>
\write9<1499 4053 C>
\write9<3821 2590 C>
\write9<3532 2048 C>
\write9<4109 3132 C>
\write9<4116 3307 C>
\write9<3520 2024 C>
\write9<4121 3158 C>
\write9<3499 1978 C>
\write9<4143 3214 C>
\write9<3460 1896 C>
\write9<4193 3434 C>
\write9<3381 1735 C>
\write9<4277 3235 C>
\write9<3031 1273 C>
\write9<2978 3065 C>
\write9<4666 2122 C>
\write9<2931 3109 C>
\write9<4720 2094 C>
\write9<2826 3206 C>
\write9<4839 2024 C>
\write9<2553 3490 C>
\write9<5154 1832 C>
\write9<5550 4453 C>
\write9<5712 3163 C>
\write9<1903 2083 C>
\write9<4032 3490 C>
\write9<4077 3862 C>
\write9<4107 3131 C>
\write9<3531 2049 C>
\write9<4113 3154 C>
\write9<4120 3197 C>
\write9<4123 3272 C>
\write9<3517 2027 C>
\write9<3489 1986 C>
\write9<3435 1916 C>
\write9<3310 1800 C>
\write9<3028 1695 C>
\write9<2977 2635 C>
\write9<4666 2117 C>
\write9<2972 3065 C>
\write9<4725 2079 C>
\write9<2903 3111 C>
\write9<4855 1995 C>
\write9<2743 3229 C>
\write9<5185 1796 C>
\write9<2262 3726 C>
\write9<5683 2979 C>
\write9<1834 2129 C>
\write9<3821 2590 C>
\write9<3769 4344 C>
\write9<4030 3513 C>
\write9<4035 3168 C>
\write9<4107 3134 C>
\write9<4109 3168 C>
\write9<3530 2048 C>
\write9<4124 3243 C>
\write9<3514 2024 C>
\write9<3480 1977 C>
\write9<3407 1895 C>
\write9<4313 3285 C>
\write9<3209 1776 C>
\write9<2977 2635 C>
\write9<4670 2116 C>
\write9<2969 3067 C>
\write9<4742 2073 C>
\write9<2888 3126 C>
\write9<2886 1715 C>
\write9<4918 1976 C>
\write9<2674 3299 C>
\write9<5549 1748 C>
\write9<5818 3009 C>
\write9<1751 1798 C>
\write9
\write9<6486 4486 M>
\write9<(Set 2) Rshow>
\write9<6654 4486 B>
\write9<3535 2055 B>
\write9<3515 2004 B>
\write9<2991 3052 B>
\write9<4651 2128 B>
\write9<4106 3125 B>
\write9<3535 2056 B>
\write9<2991 3053 B>
\write9<4652 2128 B>
\write9<3536 2056 B>
\write9<4106 3124 B>
\write9<4651 2127 B>
\write9<4106 3124 B>
\write9<4650 2127 B>
\write9<2990 3052 B>
\write9<4106 3125 B>
\write9<4131 3178 B>
\write9<4650 2128 B>
\write9<2990 3052 B>
\write9<4106 3125 B>
\write9<4126 3176 B>
\write9<4650 2128 B>
\write9<2990 3052 B>
\write9<3535 2055 B>
\write9<4106 3124 B>
\write9<4650 2127 B>
\write9<2989 3052 B>
\write9<4105 3124 B>
\write9<3535 2056 B>
\write9<2990 3053 B>
\write9<3535 2056 B>
\write9<2991 3053 B>
\write9<4651 2128 B>
\write9<3535 2055 B>
\write9<3510 2002 B>
\write9<2991 3052 B>
\write9<4651 2128 B>
\write9
\write9
\write9
\write9
\write9<%%Trailer>
\write9<%%Pages: 1>
\closeout9
\openout10=2c.ps
\write10<%!PS-Adobe-2.0>
\write10<%%Creator: gnuplot>
\write10<%%DocumentFonts: Helvetica>
\write10<%%BoundingBox: 50 50 554 770>
\write10<%%Pages: (atend)>
\write10<%%EndComments>
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10< 0 vshift R show } def>
\write10
\write10< dup stringwidth pop neg vshift R show } def>
\write10
\write10< dup stringwidth pop -2 div vshift R show } def>
\write10
\write10< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10
\write10< currentlinewidth 2 div sub M>
\write10< 0 currentlinewidth V stroke } def>
\write10
\write10< hpt neg vpt neg V hpt vpt neg V>
\write10< hpt vpt V hpt neg vpt V closepath stroke>
\write10< P } def>
\write10
\write10< currentpoint stroke M>
\write10< hpt neg vpt neg R hpt2 0 V stroke>
\write10< } def>
\write10
\write10< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write10< hpt4 neg 0 V closepath stroke>
\write10< P } def>
\write10
\write10< hpt2 vpt2 neg V currentpoint stroke M>
\write10< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write10
\write10< hpt neg vpt -1.62 mul V>
\write10< hpt 2 mul 0 V>
\write10< hpt neg vpt 1.62 mul V closepath stroke>
\write10< P } def>
\write10
\write10
\write10<%%EndProlog>
\write10<%%Page: 1 1>
\write10
\write10
\write10<50 50 translate>
\write10<0.100 0.100 scale>
\write10<90 rotate>
\write10<0 -5040 translate>
\write10<0 setgray>
\write10
\write10
\write10
\write10<672 2590 M>
\write10<6297 0 V>
\write10<3821 351 M>
\write10<0 4478 V>
\write10
\write10<672 351 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<588 351 M>
\write10<(-4) Rshow>
\write10<672 911 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<588 911 M>
\write10<(-3) Rshow>
\write10<672 1471 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(-2) Rshow>
\write10<672 2030 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(-1) Rshow>
\write10<672 2590 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(0) Rshow>
\write10<672 3150 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(1) Rshow>
\write10<672 3710 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(2) Rshow>
\write10<672 4269 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(3) Rshow>
\write10<672 4829 M>
\write10<63 0 V>
\write10<6234 0 R>
\write10<-63 0 V>
\write10<-6318 0 R>
\write10<(4) Rshow>
\write10<672 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<672 211 M>
\write10<(-4) Cshow>
\write10<1459 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(-3) Cshow>
\write10<2246 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(-2) Cshow>
\write10<3033 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(-1) Cshow>
\write10<3821 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(0) Cshow>
\write10<4608 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(1) Cshow>
\write10<5395 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(2) Cshow>
\write10<6182 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(3) Cshow>
\write10<6969 351 M>
\write10<0 63 V>
\write10<0 4415 R>
\write10<0 -63 V>
\write10<0 -4555 R>
\write10<(4) Cshow>
\write10<672 351 M>
\write10<6297 0 V>
\write10<0 4478 V>
\write10<-6297 0 V>
\write10<672 351 L>
\write10<3820 71 M>
\write10<(Im epsilon) Cshow>
\write10<3820 4969 M>
\write10<(Figure 2c) Cshow>
\write10
\write10<6486 4626 M>
\write10<(Set 1) Rshow>
\write10<6654 4626 C>
\write10<3249 1453 C>
\write10<4396 3715 C>
\write10<3211 1375 C>
\write10<4448 3831 C>
\write10<3177 1202 C>
\write10<4524 3995 C>
\write10<4590 4689 C>
\write10<2970 528 C>
\write10<5004 1803 C>
\write10<2636 3376 C>
\write10<5075 1752 C>
\write10<2563 3422 C>
\write10<5229 1640 C>
\write10<2405 3519 C>
\write10<5589 1353 C>
\write10<2035 3728 C>
\write10<5613 3045 C>
\write10<2027 2133 C>
\write10<1830 2076 C>
\write10<5813 3093 C>
\write10<1202 1887 C>
\write10<6536 3156 C>
\write10<3821 2590 C>
\write10<3247 1445 C>
\write10<4396 3714 C>
\write10<3195 1326 C>
\write10<4449 3843 C>
\write10<3192 1046 C>
\write10<4541 3430 C>
\write10<4565 4060 C>
\write10<2818 1354 C>
\write10<2632 3379 C>
\write10<5009 1801 C>
\write10<2545 3436 C>
\write10<5097 1741 C>
\write10<2343 3564 C>
\write10<5301 1601 C>
\write10<2018 2127 C>
\write10<5624 3053 C>
\write10<1768 2032 C>
\write10<5878 3146 C>
\write10<5933 1074 C>
\write10<1702 3898 C>
\write10<3250 1454 C>
\write10<4392 3724 C>
\write10<3207 1377 C>
\write10<4434 3792 C>
\write10<4518 3954 C>
\write10<3119 1197 C>
\write10<3007 615 C>
\write10<5003 1803 C>
\write10<2637 3376 C>
\write10<5068 1751 C>
\write10<2568 3421 C>
\write10<5191 1635 C>
\write10<2423 3519 C>
\write10<5350 1437 C>
\write10<2102 3743 C>
\write10<5609 3043 C>
\write10<2014 2133 C>
\write10<5782 3082 C>
\write10<1832 2288 C>
\write10<1707 2034 C>
\write10<6233 3186 C>
\write10<3821 2590 C>
\write10<4387 3724 C>
\write10<3248 1451 C>
\write10<4415 3795 C>
\write10<3196 1359 C>
\write10<4483 3964 C>
\write10<3077 1132 C>
\write10<2860 4789 C>
\write10<2637 3379 C>
\write10<5006 1803 C>
\write10<2569 3437 C>
\write10<5079 1752 C>
\write10<2433 3567 C>
\write10<5218 1641 C>
\write10<5370 1454 C>
\write10<2192 3903 C>
\write10<5602 3046 C>
\write10<2012 2130 C>
\write10<5746 3088 C>
\write10<1832 2288 C>
\write10<1716 2009 C>
\write10<6114 3181 C>
\write10<3248 1453 C>
\write10<4393 3726 C>
\write10<3200 1375 C>
\write10<4445 3804 C>
\write10<3103 1186 C>
\write10<4565 3986 C>
\write10<2643 3376 C>
\write10<5001 1804 C>
\write10<2593 3423 C>
\write10<5061 1758 C>
\write10<2484 3523 C>
\write10<5183 1656 C>
\write10<5381 4506 C>
\write10<2232 3753 C>
\write10<5420 1391 C>
\write10<2035 2129 C>
\write10<5611 3050 C>
\write10<1880 2053 C>
\write10<5803 3121 C>
\write10<1529 1830 C>
\write10<6572 3533 C>
\write10<3821 2590 C>
\write10<3249 1450 C>
\write10<4394 3730 C>
\write10<3205 1358 C>
\write10<4449 3823 C>
\write10<3130 1128 C>
\write10<4586 4061 C>
\write10<4994 1804 C>
\write10<2645 3374 C>
\write10<5038 1761 C>
\write10<2597 3414 C>
\write10<5146 1670 C>
\write10<2487 3501 C>
\write10<5387 1437 C>
\write10<2211 3689 C>
\write10<2028 2129 C>
\write10<5619 3049 C>
\write10<1841 2053 C>
\write10<5845 3114 C>
\write10<6127 4366 C>
\write10<1365 1834 C>
\write10<982 3905 C>
\write10<3251 1455 C>
\write10<4391 3726 C>
\write10<3221 1378 C>
\write10<4430 3801 C>
\write10<3162 1178 C>
\write10<4502 3978 C>
\write10<4568 4512 C>
\write10<2637 3376 C>
\write10<5004 1804 C>
\write10<2569 3426 C>
\write10<5073 1757 C>
\write10<2426 3532 C>
\write10<5218 1655 C>
\write10<2122 3801 C>
\write10<5544 1419 C>
\write10<2031 2134 C>
\write10<5611 3044 C>
\write10<1850 2082 C>
\write10<5791 3087 C>
\write10<6246 3204 C>
\write10<1307 1937 C>
\write10<3821 2590 C>
\write10<4297 440 C>
\write10<3261 1462 C>
\write10<4394 3730 C>
\write10<3241 1399 C>
\write10<4449 3820 C>
\write10<3173 1223 C>
\write10<4575 4039 C>
\write10<5006 1801 C>
\write10<2634 3377 C>
\write10<5083 1742 C>
\write10<2554 3430 C>
\write10<5248 1606 C>
\write10<2383 3543 C>
\write10<2027 3833 C>
\write10<2025 2121 C>
\write10<5616 3050 C>
\write10<5639 1181 C>
\write10<1890 1849 C>
\write10<5813 3122 C>
\write10<1814 1959 C>
\write10<6197 3375 C>
\write10<4391 3725 C>
\write10<3238 1456 C>
\write10<4430 3791 C>
\write10<3161 1379 C>
\write10<4490 4829 C>
\write10<3138 391 C>
\write10<4518 3936 C>
\write10<3060 1179 C>
\write10<2637 3376 C>
\write10<5005 1804 C>
\write10<2567 3422 C>
\write10<5077 1759 C>
\write10<2414 3516 C>
\write10<5232 1666 C>
\write10<5591 1471 C>
\write10<2030 2137 C>
\write10<5611 3048 C>
\write10<2018 3697 C>
\write10<5795 3111 C>
\write10<1841 2099 C>
\write10<6264 3341 C>
\write10<1144 3371 C>
\write10<1044 2124 C>
\write10<3821 2590 C>
\write10<4391 3733 C>
\write10<3243 1456 C>
\write10<4429 3838 C>
\write10<3184 1858 C>
\write10<4463 4051 C>
\write10<3170 1368 C>
\write10<3047 1118 C>
\write10<4733 3689 C>
\write10<2632 3378 C>
\write10<5009 1802 C>
\write10<5096 1746 C>
\write10<2544 3432 C>
\write10<5298 1624 C>
\write10<2334 3546 C>
\write10<2021 2131 C>
\write10<5621 3054 C>
\write10<1793 2062 C>
\write10<5850 3153 C>
\write10<5935 1345 C>
\write10<1540 3687 C>
\write10<6489 3607 C>
\write10<1102 2951 C>
\write10<4392 3727 C>
\write10<3245 1465 C>
\write10<4430 3805 C>
\write10<3193 1349 C>
\write10<4464 3978 C>
\write10<3117 1185 C>
\write10<3051 491 C>
\write10<4671 4652 C>
\write10<2637 3377 C>
\write10<5005 1804 C>
\write10<2566 3428 C>
\write10<5078 1758 C>
\write10<2412 3540 C>
\write10<5236 1661 C>
\write10<2052 3827 C>
\write10<5606 1452 C>
\write10<2028 2135 C>
\write10<5614 3047 C>
\write10<5811 3104 C>
\write10<1828 2087 C>
\write10<6439 3293 C>
\write10<1105 2024 C>
\write10<3821 2590 C>
\write10<4394 3735 C>
\write10<3245 1466 C>
\write10<4446 3854 C>
\write10<3192 1337 C>
\write10<4449 4134 C>
\write10<3100 1750 C>
\write10<3076 1120 C>
\write10<4823 3826 C>
\write10<5009 1801 C>
\write10<2632 3379 C>
\write10<5096 1744 C>
\write10<2544 3439 C>
\write10<5298 1616 C>
\write10<2340 3579 C>
\write10<5623 3053 C>
\write10<2017 2127 C>
\write10<5873 3148 C>
\write10<1763 2034 C>
\write10<1708 4106 C>
\write10<5939 1282 C>
\write10<4391 3726 C>
\write10<3249 1456 C>
\write10<4434 3803 C>
\write10<3207 1388 C>
\write10<3123 1226 C>
\write10<4522 3983 C>
\write10<4634 4565 C>
\write10<2638 3377 C>
\write10<5004 1804 C>
\write10<2573 3429 C>
\write10<5073 1759 C>
\write10<2450 3545 C>
\write10<5218 1661 C>
\write10<2291 3743 C>
\write10<5539 1437 C>
\write10<2032 2137 C>
\write10<5627 3047 C>
\write10<1859 2098 C>
\write10<5809 2892 C>
\write10<5934 3146 C>
\write10<1408 1994 C>
\write10<3821 2590 C>
\write10<3254 1456 C>
\write10<4393 3729 C>
\write10<3226 1385 C>
\write10<4445 3821 C>
\write10<3158 1216 C>
\write10<4564 4048 C>
\write10<4781 391 C>
\write10<5004 1801 C>
\write10<2635 3377 C>
\write10<5072 1743 C>
\write10<2562 3428 C>
\write10<5208 1613 C>
\write10<2423 3539 C>
\write10<2271 3726 C>
\write10<5449 1277 C>
\write10<2039 2134 C>
\write10<5629 3050 C>
\write10<1895 2092 C>
\write10<5809 2892 C>
\write10<5925 3171 C>
\write10<1527 1999 C>
\write10<4393 3727 C>
\write10<3248 1454 C>
\write10<4441 3805 C>
\write10<3196 1376 C>
\write10<4538 3994 C>
\write10<3076 1194 C>
\write10<4998 1804 C>
\write10<2640 3376 C>
\write10<5048 1757 C>
\write10<2580 3422 C>
\write10<5157 1657 C>
\write10<2458 3524 C>
\write10<2260 674 C>
\write10<5409 1427 C>
\write10<2221 3789 C>
\write10<5606 3051 C>
\write10<2030 2130 C>
\write10<5761 3127 C>
\write10<1838 2059 C>
\write10<6112 3350 C>
\write10<1069 1647 C>
\write10<3821 2590 C>
\write10<4392 3730 C>
\write10<3247 1450 C>
\write10<4436 3822 C>
\write10<3192 1357 C>
\write10<4511 4052 C>
\write10<3055 1119 C>
\write10<2647 3376 C>
\write10<4996 1806 C>
\write10<2603 3419 C>
\write10<5044 1766 C>
\write10<2495 3510 C>
\write10<5154 1679 C>
\write10<2254 3743 C>
\write10<5430 1491 C>
\write10<5613 3051 C>
\write10<2022 2131 C>
\write10<5800 3127 C>
\write10<1796 2066 C>
\write10<1514 814 C>
\write10<6276 3346 C>
\write10<6659 1275 C>
\write10<4390 3725 C>
\write10<3250 1454 C>
\write10<4420 3802 C>
\write10<3211 1379 C>
\write10<4479 4002 C>
\write10<3139 1202 C>
\write10<3073 668 C>
\write10<5004 1804 C>
\write10<2637 3376 C>
\write10<5072 1754 C>
\write10<2568 3423 C>
\write10<5215 1648 C>
\write10<2423 3525 C>
\write10<5519 1379 C>
\write10<2097 3761 C>
\write10<5610 3046 C>
\write10<2030 2136 C>
\write10<5791 3098 C>
\write10<1850 2093 C>
\write10<1395 1976 C>
\write10<6334 3243 C>
\write10<3821 2590 C>
\write10<3344 4740 C>
\write10<4380 3718 C>
\write10<3247 1450 C>
\write10<4400 3781 C>
\write10<3192 1360 C>
\write10<4468 3957 C>
\write10<3066 1141 C>
\write10<2635 3379 C>
\write10<5007 1803 C>
\write10<2558 3438 C>
\write10<5087 1750 C>
\write10<2393 3574 C>
\write10<5258 1637 C>
\write10<5614 1347 C>
\write10<5616 3059 C>
\write10<2025 2130 C>
\write10<2002 3999 C>
\write10<5751 3331 C>
\write10<1828 2058 C>
\write10<5827 3221 C>
\write10<1444 1805 C>
\write10<3250 1455 C>
\write10<4403 3724 C>
\write10<3211 1389 C>
\write10<4480 3801 C>
\write10<3151 351 C>
\write10<4503 4789 C>
\write10<3123 1244 C>
\write10<4581 4001 C>
\write10<5004 1804 C>
\write10<2636 3376 C>
\write10<5074 1758 C>
\write10<2564 3421 C>
\write10<5227 1664 C>
\write10<2409 3514 C>
\write10<2050 3709 C>
\write10<5611 3043 C>
\write10<2030 2132 C>
\write10<5623 1483 C>
\write10<1846 2069 C>
\write10<5800 3081 C>
\write10<1377 1839 C>
\write10<6497 1809 C>
\write10<6597 3056 C>
\write10<3821 2590 C>
\write10<3250 1447 C>
\write10<4398 3724 C>
\write10<3212 1342 C>
\write10<4457 3322 C>
\write10<3178 1129 C>
\write10<4471 3812 C>
\write10<4594 4062 C>
\write10<2908 1491 C>
\write10<5009 1802 C>
\write10<2632 3378 C>
\write10<2545 3434 C>
\write10<5097 1748 C>
\write10<2343 3556 C>
\write10<5307 1634 C>
\write10<5620 3049 C>
\write10<2020 2126 C>
\write10<5848 3118 C>
\write10<1791 2027 C>
\write10<1706 3835 C>
\write10<6101 1493 C>
\write10<1152 1573 C>
\write10<6539 2229 C>
\write10
\write10<6486 4486 M>
\write10<(Set 2) Rshow>
\write10<6654 4486 B>
\write10<4986 1816 B>
\write10<3261 1475 B>
\write10<4378 3706 B>
\write10<2078 2137 B>
\write10<5568 3032 B>
\write10<4984 1817 B>
\write10<5521 3004 B>
\write10<2654 3363 B>
\write10<5471 3035 B>
\write10<2655 3364 B>
\write10<4380 3705 B>
\write10<3263 1474 B>
\write10<5563 3043 B>
\write10<2073 2148 B>
\write10<2657 3363 B>
\write10<2120 2176 B>
\write10<4987 1817 B>
\write10<2170 2145 B>
\write10
\write10
\write10
\write10
\write10<%%Trailer>
\write10<%%Pages: 1>
\closeout10
\openout11=3a.ps
\write11<%!PS-Adobe-2.0>
\write11<%%Creator: gnuplot>
\write11<%%DocumentFonts: Helvetica>
\write11<%%BoundingBox: 50 50 554 770>
\write11<%%Pages: (atend)>
\write11<%%EndComments>
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11< 0 vshift R show } def>
\write11
\write11< dup stringwidth pop neg vshift R show } def>
\write11
\write11< dup stringwidth pop -2 div vshift R show } def>
\write11
\write11< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11
\write11< currentlinewidth 2 div sub M>
\write11< 0 currentlinewidth V stroke } def>
\write11
\write11< hpt neg vpt neg V hpt vpt neg V>
\write11< hpt vpt V hpt neg vpt V closepath stroke>
\write11< P } def>
\write11
\write11< currentpoint stroke M>
\write11< hpt neg vpt neg R hpt2 0 V stroke>
\write11< } def>
\write11
\write11< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write11< hpt4 neg 0 V closepath stroke>
\write11< P } def>
\write11
\write11< hpt2 vpt2 neg V currentpoint stroke M>
\write11< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write11
\write11< hpt neg vpt -1.62 mul V>
\write11< hpt 2 mul 0 V>
\write11< hpt neg vpt 1.62 mul V closepath stroke>
\write11< P } def>
\write11
\write11
\write11<%%EndProlog>
\write11<%%Page: 1 1>
\write11
\write11
\write11<50 50 translate>
\write11<0.100 0.100 scale>
\write11<90 rotate>
\write11<0 -5040 translate>
\write11<0 setgray>
\write11
\write11
\write11
\write11<672 2590 M>
\write11<6297 0 V>
\write11<3821 351 M>
\write11<0 4478 V>
\write11
\write11<672 799 M>
\write11<63 0 V>
\write11<6234 0 R>
\write11<-63 0 V>
\write11<588 799 M>
\write11<(-40) Rshow>
\write11<672 1694 M>
\write11<63 0 V>
\write11<6234 0 R>
\write11<-63 0 V>
\write11<-6318 0 R>
\write11<(-20) Rshow>
\write11<672 2590 M>
\write11<63 0 V>
\write11<6234 0 R>
\write11<-63 0 V>
\write11<-6318 0 R>
\write11<(0) Rshow>
\write11<672 3486 M>
\write11<63 0 V>
\write11<6234 0 R>
\write11<-63 0 V>
\write11<-6318 0 R>
\write11<(20) Rshow>
\write11<672 4381 M>
\write11<63 0 V>
\write11<6234 0 R>
\write11<-63 0 V>
\write11<-6318 0 R>
\write11<(40) Rshow>
\write11<1302 351 M>
\write11<0 63 V>
\write11<0 4415 R>
\write11<0 -63 V>
\write11<0 -4555 R>
\write11<(-40) Cshow>
\write11<2561 351 M>
\write11<0 63 V>
\write11<0 4415 R>
\write11<0 -63 V>
\write11<0 -4555 R>
\write11<(-20) Cshow>
\write11<3821 351 M>
\write11<0 63 V>
\write11<0 4415 R>
\write11<0 -63 V>
\write11<0 -4555 R>
\write11<(0) Cshow>
\write11<5080 351 M>
\write11<0 63 V>
\write11<0 4415 R>
\write11<0 -63 V>
\write11<0 -4555 R>
\write11<(20) Cshow>
\write11<6339 351 M>
\write11<0 63 V>
\write11<0 4415 R>
\write11<0 -63 V>
\write11<0 -4555 R>
\write11<(40) Cshow>
\write11<672 351 M>
\write11<6297 0 V>
\write11<0 4478 V>
\write11<-6297 0 V>
\write11<672 351 L>
\write11<3820 71 M>
\write11<(Im epsilon) Cshow>
\write11<3820 4969 M>
\write11<(Figure 3a) Cshow>
\write11
\write11<6486 4626 M>
\write11<(Set 1) Rshow>
\write11<6654 4626 C>
\write11<2911 1800 C>
\write11<4731 3382 C>
\write11<2878 1743 C>
\write11<4783 3424 C>
\write11<4783 3428 C>
\write11<2839 1766 C>
\write11<4860 3495 C>
\write11<2777 1681 C>
\write11<2691 1607 C>
\write11<4951 3574 C>
\write11<5073 3684 C>
\write11<2566 1496 C>
\write11<5256 3847 C>
\write11<2383 1334 C>
\write11<5531 4092 C>
\write11<2107 1088 C>
\write11<5976 4491 C>
\write11<1659 689 C>
\write11<3821 2590 C>
\write11<2909 1798 C>
\write11<4734 3383 C>
\write11<4776 3419 C>
\write11<2856 1754 C>
\write11<2854 1748 C>
\write11<4794 3438 C>
\write11<2769 1674 C>
\write11<4874 3506 C>
\write11<4967 3589 C>
\write11<2672 1590 C>
\write11<2534 1467 C>
\write11<5107 3713 C>
\write11<2327 1283 C>
\write11<5314 3898 C>
\write11<2004 995 C>
\write11<5637 4186 C>
\write11<1451 501 C>
\write11<6192 4680 C>
\write11<2913 1801 C>
\write11<4731 3381 C>
\write11<2890 1745 C>
\write11<4758 3428 C>
\write11<4806 3415 C>
\write11<2830 1772 C>
\write11<4852 3499 C>
\write11<2784 1677 C>
\write11<2697 1619 C>
\write11<4946 3562 C>
\write11<5062 3672 C>
\write11<2579 1508 C>
\write11<5240 3826 C>
\write11<2401 1354 C>
\write11<5509 4060 C>
\write11<2132 1119 C>
\write11<5946 4442 C>
\write11<1694 737 C>
\write11<3821 2590 C>
\write11<2910 1799 C>
\write11<4733 3383 C>
\write11<2874 1748 C>
\write11<4777 3421 C>
\write11<4790 3433 C>
\write11<2839 1760 C>
\write11<2776 1677 C>
\write11<4868 3500 C>
\write11<4959 3580 C>
\write11<2680 1601 C>
\write11<2548 1482 C>
\write11<5094 3698 C>
\write11<2346 1306 C>
\write11<5296 3874 C>
\write11<2032 1031 C>
\write11<5611 4149 C>
\write11<1491 559 C>
\write11<6154 4621 C>
\write11<4729 3380 C>
\write11<2910 1799 C>
\write11<4762 3437 C>
\write11<2859 1753 C>
\write11<2858 1755 C>
\write11<4803 3414 C>
\write11<2783 1686 C>
\write11<4862 3498 C>
\write11<4949 3572 C>
\write11<2691 1607 C>
\write11<2570 1499 C>
\write11<5071 3681 C>
\write11<2388 1338 C>
\write11<5253 3842 C>
\write11<2114 1095 C>
\write11<5527 4085 C>
\write11<1669 700 C>
\write11<5973 4480 C>
\write11<3821 2590 C>
\write11<4732 3382 C>
\write11<2907 1797 C>
\write11<2865 1761 C>
\write11<4786 3427 C>
\write11<4786 3431 C>
\write11<2848 1743 C>
\write11<4870 3504 C>
\write11<2769 1675 C>
\write11<2676 1593 C>
\write11<4967 3588 C>
\write11<5104 3710 C>
\write11<2537 1470 C>
\write11<5310 3893 C>
\write11<2331 1287 C>
\write11<5631 4178 C>
\write11<2010 1002 C>
\write11<6182 4668 C>
\write11<1459 513 C>
\write11<4729 3380 C>
\write11<2910 1799 C>
\write11<4761 3437 C>
\write11<2860 1753 C>
\write11<2856 1755 C>
\write11<4803 3413 C>
\write11<2785 1686 C>
\write11<4860 3498 C>
\write11<4947 3571 C>
\write11<2694 1607 C>
\write11<2574 1499 C>
\write11<5067 3681 C>
\write11<2395 1339 C>
\write11<5247 3842 C>
\write11<2125 1097 C>
\write11<5517 4083 C>
\write11<1685 704 C>
\write11<5957 4476 C>
\write11<3821 2590 C>
\write11<4732 3382 C>
\write11<2908 1797 C>
\write11<2865 1761 C>
\write11<4784 3431 C>
\write11<4787 3426 C>
\write11<2849 1743 C>
\write11<4868 3505 C>
\write11<2770 1675 C>
\write11<2678 1593 C>
\write11<4964 3588 C>
\write11<5098 3710 C>
\write11<2542 1470 C>
\write11<5302 3892 C>
\write11<2339 1288 C>
\write11<5618 4175 C>
\write11<2021 1005 C>
\write11<6161 4661 C>
\write11<1475 519 C>
\write11<4729 3380 C>
\write11<2910 1798 C>
\write11<4763 3438 C>
\write11<2858 1757 C>
\write11<2858 1750 C>
\write11<4801 3414 C>
\write11<2781 1682 C>
\write11<4864 3501 C>
\write11<4949 3576 C>
\write11<2691 1603 C>
\write11<2568 1491 C>
\write11<5074 3689 C>
\write11<2386 1326 C>
\write11<5256 3855 C>
\write11<2111 1076 C>
\write11<5531 4106 C>
\write11<1666 669 C>
\write11<5976 4515 C>
\write11<3821 2590 C>
\write11<4732 3383 C>
\write11<2907 1796 C>
\write11<2866 1762 C>
\write11<4785 3424 C>
\write11<4787 3434 C>
\write11<2847 1741 C>
\write11<4872 3508 C>
\write11<2768 1672 C>
\write11<2674 1588 C>
\write11<4968 3593 C>
\write11<5106 3719 C>
\write11<2534 1461 C>
\write11<5312 3907 C>
\write11<2327 1274 C>
\write11<5633 4203 C>
\write11<2004 980 C>
\write11<6181 4709 C>
\write11<1450 476 C>
\write11<4730 3380 C>
\write11<2909 1798 C>
\write11<4764 3437 C>
\write11<2860 1757 C>
\write11<2855 1751 C>
\write11<4802 3415 C>
\write11<2778 1683 C>
\write11<4867 3500 C>
\write11<4953 3575 C>
\write11<2687 1604 C>
\write11<2561 1492 C>
\write11<5082 3688 C>
\write11<2373 1327 C>
\write11<5270 3853 C>
\write11<2089 1077 C>
\write11<5556 4104 C>
\write11<1627 669 C>
\write11<6022 4512 C>
\write11<3821 2590 C>
\write11<4733 3382 C>
\write11<2907 1796 C>
\write11<2866 1761 C>
\write11<4783 3425 C>
\write11<4790 3433 C>
\write11<2844 1741 C>
\write11<4875 3507 C>
\write11<2765 1672 C>
\write11<2669 1589 C>
\write11<4972 3592 C>
\write11<5115 3717 C>
\write11<2526 1463 C>
\write11<2312 1275 C>
\write11<5329 3905 C>
\write11<1977 980 C>
\write11<5664 4200 C>
\write11<1401 473 C>
\write11<6241 4707 C>
\write11<4729 3380 C>
\write11<2910 1799 C>
\write11<4762 3437 C>
\write11<2859 1753 C>
\write11<2858 1755 C>
\write11<4803 3414 C>
\write11<2783 1686 C>
\write11<4862 3498 C>
\write11<4949 3572 C>
\write11<2691 1607 C>
\write11<2570 1498 C>
\write11<5071 3682 C>
\write11<2388 1336 C>
\write11<5253 3844 C>
\write11<2115 1093 C>
\write11<5526 4087 C>
\write11<1672 698 C>
\write11<5970 4483 C>
\write11<3821 2590 C>
\write11<4732 3382 C>
\write11<2907 1797 C>
\write11<2865 1761 C>
\write11<4786 3426 C>
\write11<4786 3431 C>
\write11<2847 1743 C>
\write11<4870 3505 C>
\write11<2769 1674 C>
\write11<2676 1592 C>
\write11<4967 3589 C>
\write11<5103 3711 C>
\write11<2537 1469 C>
\write11<5309 3894 C>
\write11<2331 1286 C>
\write11<5629 4180 C>
\write11<2011 1000 C>
\write11<6177 4670 C>
\write11<1460 510 C>
\write11<2912 1801 C>
\write11<4731 3381 C>
\write11<2880 1742 C>
\write11<4780 3426 C>
\write11<4785 3426 C>
\write11<2839 1767 C>
\write11<4855 3492 C>
\write11<2780 1685 C>
\write11<2695 1609 C>
\write11<4947 3571 C>
\write11<5066 3678 C>
\write11<2574 1502 C>
\write11<5247 3837 C>
\write11<2394 1343 C>
\write11<5517 4078 C>
\write11<2123 1102 C>
\write11<5958 4468 C>
\write11<1683 712 C>
\write11<3821 2590 C>
\write11<2910 1798 C>
\write11<4733 3383 C>
\write11<4776 3420 C>
\write11<2857 1751 C>
\write11<2854 1752 C>
\write11<4792 3436 C>
\write11<2774 1677 C>
\write11<4870 3504 C>
\write11<4962 3585 C>
\write11<2677 1593 C>
\write11<2543 1474 C>
\write11<5099 3707 C>
\write11<2339 1293 C>
\write11<5302 3887 C>
\write11<2021 1011 C>
\write11<5620 4169 C>
\write11<1476 528 C>
\write11<6165 4653 C>
\write11<2911 1800 C>
\write11<4731 3381 C>
\write11<2879 1743 C>
\write11<4783 3425 C>
\write11<4783 3426 C>
\write11<2838 1766 C>
\write11<4860 3493 C>
\write11<2777 1683 C>
\write11<2691 1609 C>
\write11<4951 3572 C>
\write11<5074 3679 C>
\write11<2566 1500 C>
\write11<5257 3839 C>
\write11<2383 1341 C>
\write11<5533 4080 C>
\write11<2106 1100 C>
\write11<5982 4471 C>
\write11<1656 708 C>
\write11<3821 2590 C>
\write11<2909 1798 C>
\write11<4734 3383 C>
\write11<4775 3419 C>
\write11<2856 1753 C>
\write11<2854 1750 C>
\write11<4795 3436 C>
\write11<2769 1677 C>
\write11<4874 3505 C>
\write11<4967 3586 C>
\write11<2672 1593 C>
\write11<2534 1472 C>
\write11<5108 3708 C>
\write11<2326 1291 C>
\write11<5315 3889 C>
\write11<2002 1008 C>
\write11<5640 4171 C>
\write11<1447 523 C>
\write11<6198 4655 C>
\write11<2912 1799 C>
\write11<4731 3383 C>
\write11<2879 1741 C>
\write11<4782 3420 C>
\write11<4783 3434 C>
\write11<2839 1767 C>
\write11<4860 3503 C>
\write11<2774 1671 C>
\write11<2694 1599 C>
\write11<4949 3583 C>
\write11<5070 3703 C>
\write11<2570 1476 C>
\write11<5246 3879 C>
\write11<2394 1300 C>
\write11<5502 4147 C>
\write11<2136 1030 C>
\write11<5902 4580 C>
\write11<1737 595 C>
\write11<3821 2590 C>
\write11<2909 1797 C>
\write11<4734 3385 C>
\write11<4776 3417 C>
\write11<2856 1759 C>
\write11<2854 1740 C>
\write11<4794 3443 C>
\write11<2768 1666 C>
\write11<4873 3513 C>
\write11<4965 3601 C>
\write11<2675 1579 C>
\write11<2539 1444 C>
\write11<5102 3735 C>
\write11<2341 1243 C>
\write11<5300 3936 C>
\write11<2045 925 C>
\write11<5598 4253 C>
\write11<1561 388 C>
\write11<6088 4787 C>
\write11
\write11<6486 4486 M>
\write11<(Set 2) Rshow>
\write11<6654 4486 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11<2911 1800 B>
\write11<2911 1800 B>
\write11<4730 3380 B>
\write11
\write11
\write11
\write11
\write11<%%Trailer>
\write11<%%Pages: 1>
\closeout11
\openout12=3b.ps
\write12<%!PS-Adobe-2.0>
\write12<%%Creator: gnuplot>
\write12<%%DocumentFonts: Helvetica>
\write12<%%BoundingBox: 50 50 554 770>
\write12<%%Pages: (atend)>
\write12<%%EndComments>
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12< 0 vshift R show } def>
\write12
\write12< dup stringwidth pop neg vshift R show } def>
\write12
\write12< dup stringwidth pop -2 div vshift R show } def>
\write12
\write12< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12
\write12< currentlinewidth 2 div sub M>
\write12< 0 currentlinewidth V stroke } def>
\write12
\write12< hpt neg vpt neg V hpt vpt neg V>
\write12< hpt vpt V hpt neg vpt V closepath stroke>
\write12< P } def>
\write12
\write12< currentpoint stroke M>
\write12< hpt neg vpt neg R hpt2 0 V stroke>
\write12< } def>
\write12
\write12< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write12< hpt4 neg 0 V closepath stroke>
\write12< P } def>
\write12
\write12< hpt2 vpt2 neg V currentpoint stroke M>
\write12< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write12
\write12< hpt neg vpt -1.62 mul V>
\write12< hpt 2 mul 0 V>
\write12< hpt neg vpt 1.62 mul V closepath stroke>
\write12< P } def>
\write12
\write12
\write12<%%EndProlog>
\write12<%%Page: 1 1>
\write12
\write12
\write12<50 50 translate>
\write12<0.100 0.100 scale>
\write12<90 rotate>
\write12<0 -5040 translate>
\write12<0 setgray>
\write12
\write12
\write12
\write12<672 2590 M>
\write12<6297 0 V>
\write12<3821 351 M>
\write12<0 4478 V>
\write12
\write12<672 351 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<588 351 M>
\write12<(-6) Rshow>
\write12<672 1097 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(-4) Rshow>
\write12<672 1844 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(-2) Rshow>
\write12<672 2590 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(0) Rshow>
\write12<672 3336 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(2) Rshow>
\write12<672 4083 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(4) Rshow>
\write12<672 4829 M>
\write12<63 0 V>
\write12<6234 0 R>
\write12<-63 0 V>
\write12<-6318 0 R>
\write12<(6) Rshow>
\write12<672 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<672 211 M>
\write12<(-6) Cshow>
\write12<1722 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(-4) Cshow>
\write12<2771 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(-2) Cshow>
\write12<3821 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(0) Cshow>
\write12<4870 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(2) Cshow>
\write12<5920 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(4) Cshow>
\write12<6969 351 M>
\write12<0 63 V>
\write12<0 4415 R>
\write12<0 -63 V>
\write12<0 -4555 R>
\write12<(6) Cshow>
\write12<672 351 M>
\write12<6297 0 V>
\write12<0 4478 V>
\write12<-6297 0 V>
\write12<672 351 L>
\write12<3820 71 M>
\write12<(Im epsilon) Cshow>
\write12<3820 4969 M>
\write12<(Figure 3b) Cshow>
\write12
\write12<6486 4626 M>
\write12<(Set 1) Rshow>
\write12<6654 4626 C>
\write12<4095 1475 C>
\write12<4110 1510 C>
\write12<4116 1415 C>
\write12<3523 3670 C>
\write12<3509 3719 C>
\write12<3483 3816 C>
\write12<4168 1282 C>
\write12<3431 4002 C>
\write12<4328 998 C>
\write12<3230 4498 C>
\write12<4619 4757 C>
\write12<5007 983 C>
\write12<5793 2734 C>
\write12<1846 2445 C>
\write12<5944 2751 C>
\write12<1692 1445 C>
\write12<1683 2419 C>
\write12<6171 3629 C>
\write12<6266 2802 C>
\write12<1331 4487 C>
\write12<1285 2335 C>
\write12<6938 3208 C>
\write12<3821 2590 C>
\write12<4096 1472 C>
\write12<4110 1509 C>
\write12<3524 3671 C>
\write12<4127 1406 C>
\write12<3513 3728 C>
\write12<3494 3841 C>
\write12<3471 4080 C>
\write12<4205 1259 C>
\write12<4512 973 C>
\write12<5069 1015 C>
\write12<5170 3990 C>
\write12<5800 2734 C>
\write12<1841 2444 C>
\write12<1691 1437 C>
\write12<5980 2754 C>
\write12<1654 2413 C>
\write12<1361 4588 C>
\write12<6314 3652 C>
\write12<6388 2826 C>
\write12<1161 2287 C>
\write12<3769 1483 C>
\write12<4117 1507 C>
\write12<3523 3670 C>
\write12<4126 1446 C>
\write12<3508 3721 C>
\write12<4153 1316 C>
\write12<3479 3822 C>
\write12<3416 4022 C>
\write12<4257 1006 C>
\write12<3022 4519 C>
\write12<2611 1545 C>
\write12<5102 951 C>
\write12<5475 4621 C>
\write12<2164 3902 C>
\write12<5794 2739 C>
\write12<1843 2442 C>
\write12<5953 2780 C>
\write12<1666 2401 C>
\write12<6298 2923 C>
\write12<1229 2234 C>
\write12<1208 1431 C>
\write12<6504 3623 C>
\write12<3821 2590 C>
\write12<3769 1483 C>
\write12<3523 3672 C>
\write12<4120 1508 C>
\write12<3509 3728 C>
\write12<4144 1452 C>
\write12<3482 3843 C>
\write12<4194 1343 C>
\write12<3437 4089 C>
\write12<4311 1124 C>
\write12<3188 4778 C>
\write12<5000 948 C>
\write12<2611 1545 C>
\write12<2164 3901 C>
\write12<5619 4490 C>
\write12<5799 2740 C>
\write12<1838 2442 C>
\write12<5975 2791 C>
\write12<1639 2401 C>
\write12<6251 3681 C>
\write12<6361 2988 C>
\write12<1102 2203 C>
\write12<1091 1248 C>
\write12<3545 1052 C>
\write12<4118 1508 C>
\write12<3521 3673 C>
\write12<4130 1449 C>
\write12<3502 3734 C>
\write12<4153 1325 C>
\write12<3459 3865 C>
\write12<3322 4194 C>
\write12<4359 2728 C>
\write12<4681 4022 C>
\write12<2575 1623 C>
\write12<2541 4214 C>
\write12<5125 868 C>
\write12<5479 2171 C>
\write12<1848 2441 C>
\write12<5801 2738 C>
\write12<1696 2402 C>
\write12<5986 2782 C>
\write12<1358 2285 C>
\write12<6389 2968 C>
\write12<6682 3687 C>
\write12<3821 2590 C>
\write12<3546 1051 C>
\write12<4118 1507 C>
\write12<3521 3672 C>
\write12<4130 1446 C>
\write12<3499 3732 C>
\write12<4154 1318 C>
\write12<3454 3855 C>
\write12<3333 4124 C>
\write12<4359 2728 C>
\write12<4681 4022 C>
\write12<2603 4234 C>
\write12<5060 814 C>
\write12<2575 1623 C>
\write12<5479 2171 C>
\write12<1842 2441 C>
\write12<5809 2738 C>
\write12<1663 2401 C>
\write12<6032 2781 C>
\write12<1230 2265 C>
\write12<6563 3042 C>
\write12<6893 4236 C>
\write12<3633 3773 C>
\write12<3529 3731 C>
\write12<3527 3672 C>
\write12<4119 1510 C>
\write12<4134 1460 C>
\write12<3505 3880 C>
\write12<4167 1361 C>
\write12<4236 1168 C>
\write12<3377 4182 C>
\write12<4565 710 C>
\write12<2556 4226 C>
\write12<5471 1406 C>
\write12<5640 3586 C>
\write12<1847 2444 C>
\write12<5797 2738 C>
\write12<1686 2416 C>
\write12<5971 2777 C>
\write12<1423 1486 C>
\write12<1306 2332 C>
\write12<6392 2937 C>
\write12<6673 3804 C>
\write12<3821 2590 C>
\write12<3632 3774 C>
\write12<3524 3673 C>
\write12<4118 1508 C>
\write12<3515 3741 C>
\write12<4133 1452 C>
\write12<4161 1335 C>
\write12<3439 3894 C>
\write12<4213 1084 C>
\write12<3155 4101 C>
\write12<2641 4155 C>
\write12<5471 1406 C>
\write12<5640 3586 C>
\write12<1883 1482 C>
\write12<1840 2442 C>
\write12<5802 2739 C>
\write12<1648 2402 C>
\write12<5997 2786 C>
\write12<6476 3014 C>
\write12<1143 2266 C>
\write12<869 1374 C>
\write12<6830 4192 C>
\write12<3541 3938 C>
\write12<3531 3811 C>
\write12<3526 3670 C>
\write12<3523 3720 C>
\write12<4118 1510 C>
\write12<4132 1461 C>
\write12<4161 1365 C>
\write12<3432 4217 C>
\write12<4218 1182 C>
\write12<4477 710 C>
\write12<2891 501 C>
\write12<2558 4222 C>
\write12<1850 2445 C>
\write12<5794 2736 C>
\write12<5904 3718 C>
\write12<1705 2423 C>
\write12<5956 2762 C>
\write12<1508 1556 C>
\write12<1396 2350 C>
\write12<6349 2851 C>
\write12<764 1911 C>
\write12<3821 2590 C>
\write12<3527 3828 C>
\write12<3525 3671 C>
\write12<4117 1509 C>
\write12<3520 3726 C>
\write12<3518 3964 C>
\write12<4129 1453 C>
\write12<4150 1340 C>
\write12<4181 1104 C>
\write12<3265 4296 C>
\write12<2572 1106 C>
\write12<2520 4189 C>
\write12<1844 2444 C>
\write12<5800 2737 C>
\write12<5906 3723 C>
\write12<1673 2418 C>
\write12<5986 2770 C>
\write12<1417 1531 C>
\write12<1279 2314 C>
\write12<6472 2904 C>
\write12<3546 3705 C>
\write12<3531 3670 C>
\write12<3525 3765 C>
\write12<4118 1510 C>
\write12<4132 1461 C>
\write12<4158 1364 C>
\write12<3473 3898 C>
\write12<4210 1178 C>
\write12<3313 4182 C>
\write12<4411 682 C>
\write12<3022 423 C>
\write12<2634 4197 C>
\write12<1848 2446 C>
\write12<5795 2735 C>
\write12<1697 2429 C>
\write12<5949 3735 C>
\write12<5958 2761 C>
\write12<1470 1551 C>
\write12<1375 2378 C>
\write12<6310 693 C>
\write12<6356 2845 C>
\write12<703 1972 C>
\write12<3821 2590 C>
\write12<3545 3708 C>
\write12<3531 3671 C>
\write12<4117 1509 C>
\write12<3514 3774 C>
\write12<4128 1452 C>
\write12<4147 1339 C>
\write12<4170 1100 C>
\write12<3436 3921 C>
\write12<3129 4207 C>
\write12<2572 4165 C>
\write12<2471 1190 C>
\write12<1841 2446 C>
\write12<5800 2736 C>
\write12<5950 3743 C>
\write12<1661 2426 C>
\write12<5987 2767 C>
\write12<6280 592 C>
\write12<1327 1528 C>
\write12<1253 2354 C>
\write12<6480 2893 C>
\write12<3872 3697 C>
\write12<3524 3673 C>
\write12<4118 1510 C>
\write12<3515 3734 C>
\write12<4133 1459 C>
\write12<3488 3864 C>
\write12<4162 1358 C>
\write12<4225 1158 C>
\write12<3384 4174 C>
\write12<4619 661 C>
\write12<5030 3635 C>
\write12<2539 4229 C>
\write12<2166 559 C>
\write12<5477 1278 C>
\write12<1847 2441 C>
\write12<5798 2738 C>
\write12<1688 2400 C>
\write12<5975 2779 C>
\write12<1343 2257 C>
\write12<6412 2946 C>
\write12<6433 3749 C>
\write12<1137 1557 C>
\write12<3821 2590 C>
\write12<3872 3697 C>
\write12<4118 1508 C>
\write12<3521 3672 C>
\write12<4132 1452 C>
\write12<3497 3728 C>
\write12<4159 1337 C>
\write12<3447 3837 C>
\write12<4204 1091 C>
\write12<3330 4056 C>
\write12<4453 402 C>
\write12<2641 4232 C>
\write12<5030 3635 C>
\write12<5477 1279 C>
\write12<2022 690 C>
\write12<1842 2440 C>
\write12<5803 2738 C>
\write12<1666 2389 C>
\write12<6002 2779 C>
\write12<1390 1499 C>
\write12<1280 2192 C>
\write12<6539 2977 C>
\write12<6550 3932 C>
\write12<4096 4128 C>
\write12<3523 3672 C>
\write12<4120 1507 C>
\write12<3511 3731 C>
\write12<4139 1446 C>
\write12<3488 3855 C>
\write12<4182 1315 C>
\write12<4319 986 C>
\write12<3282 2452 C>
\write12<2960 1158 C>
\write12<5066 3557 C>
\write12<5100 966 C>
\write12<2516 4312 C>
\write12<2162 3009 C>
\write12<5793 2739 C>
\write12<1840 2442 C>
\write12<5945 2778 C>
\write12<1655 2398 C>
\write12<6283 2895 C>
\write12<1252 2212 C>
\write12<959 1493 C>
\write12<3821 2590 C>
\write12<4095 4129 C>
\write12<3523 3673 C>
\write12<4120 1508 C>
\write12<3511 3734 C>
\write12<4142 1448 C>
\write12<3487 3862 C>
\write12<4187 1325 C>
\write12<4308 1056 C>
\write12<3282 2452 C>
\write12<2960 1158 C>
\write12<5038 946 C>
\write12<2581 4366 C>
\write12<5066 3557 C>
\write12<2162 3009 C>
\write12<5799 2739 C>
\write12<1832 2442 C>
\write12<5978 2779 C>
\write12<1609 2399 C>
\write12<6411 2915 C>
\write12<1078 2138 C>
\write12<748 944 C>
\write12<4008 1407 C>
\write12<4112 1449 C>
\write12<4114 1508 C>
\write12<3522 3670 C>
\write12<3507 3720 C>
\write12<4136 1300 C>
\write12<3474 3819 C>
\write12<3405 4012 C>
\write12<4264 998 C>
\write12<3076 4470 C>
\write12<5085 954 C>
\write12<2170 3774 C>
\write12<2001 1594 C>
\write12<5794 2736 C>
\write12<1844 2442 C>
\write12<5955 2764 C>
\write12<1670 2403 C>
\write12<6218 3694 C>
\write12<6335 2848 C>
\write12<1249 2243 C>
\write12<968 1376 C>
\write12<3821 2590 C>
\write12<4009 1406 C>
\write12<4117 1507 C>
\write12<3523 3672 C>
\write12<4126 1439 C>
\write12<3508 3728 C>
\write12<3480 3845 C>
\write12<4202 1286 C>
\write12<3428 4096 C>
\write12<4486 1079 C>
\write12<5000 1025 C>
\write12<2170 3774 C>
\write12<2001 1594 C>
\write12<5758 3698 C>
\write12<5801 2738 C>
\write12<1839 2441 C>
\write12<5993 2778 C>
\write12<1644 2394 C>
\write12<1165 2166 C>
\write12<6498 2914 C>
\write12<6772 3806 C>
\write12<811 988 C>
\write12<4100 1242 C>
\write12<4110 1369 C>
\write12<4115 1510 C>
\write12<4118 1460 C>
\write12<3523 3670 C>
\write12<3509 3719 C>
\write12<3480 3815 C>
\write12<4209 963 C>
\write12<3423 3998 C>
\write12<3164 4470 C>
\write12<4750 4679 C>
\write12<5083 958 C>
\write12<5791 2735 C>
\write12<1847 2444 C>
\write12<1737 1462 C>
\write12<5936 2757 C>
\write12<1685 2418 C>
\write12<6133 3624 C>
\write12<6245 2830 C>
\write12<1292 2329 C>
\write12<6877 3269 C>
\write12<3821 2590 C>
\write12<4114 1352 C>
\write12<4116 1509 C>
\write12<3524 3671 C>
\write12<4121 1454 C>
\write12<4123 1216 C>
\write12<3512 3727 C>
\write12<3491 3840 C>
\write12<3460 4076 C>
\write12<4376 884 C>
\write12<5069 4074 C>
\write12<5121 991 C>
\write12<5797 2736 C>
\write12<1841 2443 C>
\write12<1735 1457 C>
\write12<5968 2762 C>
\write12<1655 2410 C>
\write12<6224 3649 C>
\write12<6362 2866 C>
\write12<1169 2276 C>
\write12
\write12<6486 4486 M>
\write12<(Set 2) Rshow>
\write12<6654 4486 B>
\write12<3527 3655 B>
\write12<4114 1525 B>
\write12<1891 2451 B>
\write12<5752 2728 B>
\write12<3527 3655 B>
\write12<4114 1525 B>
\write12<5749 2729 B>
\write12<1892 2452 B>
\write12<4113 1525 B>
\write12<1891 2453 B>
\write12<5750 2728 B>
\write12<4114 1525 B>
\write12<3526 3655 B>
\write12<5748 2730 B>
\write12<1890 2451 B>
\write12<4114 1525 B>
\write12<3527 3655 B>
\write12<5750 2729 B>
\write12<1890 2451 B>
\write12<4114 1525 B>
\write12<3527 3655 B>
\write12<5750 2729 B>
\write12<1889 2452 B>
\write12<4114 1525 B>
\write12<3527 3655 B>
\write12<1892 2451 B>
\write12<5749 2728 B>
\write12<3528 3655 B>
\write12<5750 2727 B>
\write12<1891 2452 B>
\write12<3527 3655 B>
\write12<4115 1525 B>
\write12<1893 2450 B>
\write12<5751 2729 B>
\write12<3527 3655 B>
\write12<4114 1525 B>
\write12<1891 2451 B>
\write12<5751 2729 B>
\write12
\write12
\write12
\write12
\write12<%%Trailer>
\write12<%%Pages: 1>
\closeout12
\openout13=3c.ps
\write13<%!PS-Adobe-2.0>
\write13<%%Creator: gnuplot>
\write13<%%DocumentFonts: Helvetica>
\write13<%%BoundingBox: 50 50 554 770>
\write13<%%Pages: (atend)>
\write13<%%EndComments>
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13< 0 vshift R show } def>
\write13
\write13< dup stringwidth pop neg vshift R show } def>
\write13
\write13< dup stringwidth pop -2 div vshift R show } def>
\write13
\write13< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13
\write13< currentlinewidth 2 div sub M>
\write13< 0 currentlinewidth V stroke } def>
\write13
\write13< hpt neg vpt neg V hpt vpt neg V>
\write13< hpt vpt V hpt neg vpt V closepath stroke>
\write13< P } def>
\write13
\write13< currentpoint stroke M>
\write13< hpt neg vpt neg R hpt2 0 V stroke>
\write13< } def>
\write13
\write13< 0 vpt4 neg V hpt4 0 V 0 vpt4 V>
\write13< hpt4 neg 0 V closepath stroke>
\write13< P } def>
\write13
\write13< hpt2 vpt2 neg V currentpoint stroke M>
\write13< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write13
\write13< hpt neg vpt -1.62 mul V>
\write13< hpt 2 mul 0 V>
\write13< hpt neg vpt 1.62 mul V closepath stroke>
\write13< P } def>
\write13
\write13
\write13<%%EndProlog>
\write13<%%Page: 1 1>
\write13
\write13
\write13<50 50 translate>
\write13<0.100 0.100 scale>
\write13<90 rotate>
\write13<0 -5040 translate>
\write13<0 setgray>
\write13
\write13
\write13
\write13<672 2590 M>
\write13<6297 0 V>
\write13<3821 351 M>
\write13<0 4478 V>
\write13
\write13<672 351 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<588 351 M>
\write13<(-3) Rshow>
\write13<672 1097 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(-2) Rshow>
\write13<672 1844 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(-1) Rshow>
\write13<672 2590 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(0) Rshow>
\write13<672 3336 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(1) Rshow>
\write13<672 4083 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(2) Rshow>
\write13<672 4829 M>
\write13<63 0 V>
\write13<6234 0 R>
\write13<-63 0 V>
\write13<-6318 0 R>
\write13<(3) Rshow>
\write13<672 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<672 211 M>
\write13<(-3) Cshow>
\write13<1722 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(-2) Cshow>
\write13<2771 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(-1) Cshow>
\write13<3821 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(0) Cshow>
\write13<4870 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(1) Cshow>
\write13<5920 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(2) Cshow>
\write13<6969 351 M>
\write13<0 63 V>
\write13<0 4415 R>
\write13<0 -63 V>
\write13<0 -4555 R>
\write13<(3) Cshow>
\write13<672 351 M>
\write13<6297 0 V>
\write13<0 4478 V>
\write13<-6297 0 V>
\write13<672 351 L>
\write13<3820 71 M>
\write13<(Im epsilon) Cshow>
\write13<3820 4969 M>
\write13<(Figure 3c) Cshow>
\write13
\write13<6486 4626 M>
\write13<(Set 1) Rshow>
\write13<6654 4626 C>
\write13<3821 3804 C>
\write13<3822 1458 C>
\write13<3816 3723 C>
\write13<3834 3985 C>
\write13<3803 1381 C>
\write13<3867 4422 C>
\write13<3754 1216 C>
\write13<3582 833 C>
\write13<2951 3703 C>
\write13<2600 2354 C>
\write13<5041 2825 C>
\write13<2551 2341 C>
\write13<5092 2838 C>
\write13<2460 2314 C>
\write13<5187 2860 C>
\write13<2307 2264 C>
\write13<5349 2895 C>
\write13<5551 3005 C>
\write13<2038 2140 C>
\write13<5798 2158 C>
\write13<1538 1843 C>
\write13<6206 3212 C>
\write13<3821 2590 C>
\write13<3822 1456 C>
\write13<3818 3726 C>
\write13<3833 3819 C>
\write13<3807 1370 C>
\write13<3767 1179 C>
\write13<3879 4036 C>
\write13<3650 684 C>
\write13<4070 4660 C>
\write13<2951 3703 C>
\write13<2599 2354 C>
\write13<5043 2826 C>
\write13<2546 2339 C>
\write13<5099 2840 C>
\write13<2445 2309 C>
\write13<5209 2866 C>
\write13<2273 2250 C>
\write13<5396 2905 C>
\write13<5603 3407 C>
\write13<1938 2093 C>
\write13<5798 2158 C>
\write13<5813 3215 C>
\write13<1361 1616 C>
\write13<3820 1458 C>
\write13<3817 3731 C>
\write13<3835 3829 C>
\write13<3794 1382 C>
\write13<3764 3654 C>
\write13<3887 4034 C>
\write13<3714 1222 C>
\write13<4064 4522 C>
\write13<3448 959 C>
\write13<2600 2354 C>
\write13<5041 2826 C>
\write13<2551 2338 C>
\write13<5091 2838 C>
\write13<2458 2308 C>
\write13<5185 2862 C>
\write13<2303 2250 C>
\write13<5342 2897 C>
\write13<5530 3001 C>
\write13<2093 2091 C>
\write13<1677 1779 C>
\write13<6294 3132 C>
\write13<3821 2590 C>
\write13<3819 3736 C>
\write13<3832 1418 C>
\write13<3841 1464 C>
\write13<3848 3852 C>
\write13<3767 1290 C>
\write13<3764 3654 C>
\write13<3937 4109 C>
\write13<3537 1055 C>
\write13<2791 665 C>
\write13<2599 2353 C>
\write13<5043 2827 C>
\write13<2546 2337 C>
\write13<5098 2846 C>
\write13<5175 3070 C>
\write13<2446 2304 C>
\write13<5210 2884 C>
\write13<2275 2244 C>
\write13<5406 2932 C>
\write13<1982 2043 C>
\write13<1587 1423 C>
\write13<6139 3171 C>
\write13<3820 1458 C>
\write13<3823 3723 C>
\write13<3793 1381 C>
\write13<3858 3809 C>
\write13<3712 1217 C>
\write13<3948 4018 C>
\write13<4188 4611 C>
\write13<3024 938 C>
\write13<5041 2824 C>
\write13<2598 2356 C>
\write13<5090 2829 C>
\write13<2540 2347 C>
\write13<2496 2465 C>
\write13<5174 2835 C>
\write13<2422 2324 C>
\write13<5289 2859 C>
\write13<2210 2274 C>
\write13<5543 2938 C>
\write13<1883 2173 C>
\write13<5846 2962 C>
\write13<910 1709 C>
\write13<6749 3426 C>
\write13<3821 2590 C>
\write13<3818 1457 C>
\write13<3825 3723 C>
\write13<3782 1375 C>
\write13<3866 3808 C>
\write13<3677 1207 C>
\write13<3967 3996 C>
\write13<4189 4398 C>
\write13<3055 956 C>
\write13<5043 2824 C>
\write13<2596 2356 C>
\write13<5097 2829 C>
\write13<2530 2346 C>
\write13<2496 2465 C>
\write13<5188 2834 C>
\write13<2392 2320 C>
\write13<5323 2863 C>
\write13<2136 2266 C>
\write13<5653 2906 C>
\write13<1675 2089 C>
\write13<6086 3044 C>
\write13<3820 1457 C>
\write13<3819 3724 C>
\write13<3836 3809 C>
\write13<3797 1373 C>
\write13<3879 4001 C>
\write13<3733 1192 C>
\write13<3996 4508 C>
\write13<3628 2245 C>
\write13<3502 750 C>
\write13<2789 2657 C>
\write13<4906 3264 C>
\write13<5042 2827 C>
\write13<2598 2353 C>
\write13<5094 2843 C>
\write13<2540 2337 C>
\write13<5194 2877 C>
\write13<2431 2301 C>
\write13<5376 2941 C>
\write13<2231 2224 C>
\write13<5659 3100 C>
\write13<1931 2097 C>
\write13<1329 1412 C>
\write13<6366 3504 C>
\write13<3821 2590 C>
\write13<3820 4031 C>
\write13<3822 3821 C>
\write13<3817 1453 C>
\write13<3817 3726 C>
\write13<3797 4511 C>
\write13<3778 1356 C>
\write13<3658 1129 C>
\write13<3628 2245 C>
\write13<2881 416 C>
\write13<2789 2657 C>
\write13<4906 3264 C>
\write13<2705 1707 C>
\write13<5043 2827 C>
\write13<2595 2352 C>
\write13<5101 2846 C>
\write13<2529 2332 C>
\write13<5213 2883 C>
\write13<2398 2286 C>
\write13<5421 2953 C>
\write13<2113 2191 C>
\write13<5828 3179 C>
\write13<1664 1755 C>
\write13<6746 4329 C>
\write13<3820 3721 C>
\write13<3823 1458 C>
\write13<3812 1381 C>
\write13<3838 3798 C>
\write13<3782 1215 C>
\write13<3887 3961 C>
\write13<3688 836 C>
\write13<4053 4347 C>
\write13<2600 2355 C>
\write13<5041 2826 C>
\write13<2551 2346 C>
\write13<5091 2839 C>
\write13<5141 1936 C>
\write13<2462 2331 C>
\write13<5185 2865 C>
\write13<2339 2305 C>
\write13<5345 2916 C>
\write13<2150 2193 C>
\write13<5653 3042 C>
\write13<1734 2119 C>
\write13<6164 3456 C>
\write13<6824 4469 C>
\write13<3821 2590 C>
\write13<3821 1456 C>
\write13<3819 3724 C>
\write13<3834 3810 C>
\write13<3801 1369 C>
\write13<3873 4002 C>
\write13<3742 1178 C>
\write13<3971 4523 C>
\write13<3517 705 C>
\write13<5042 2826 C>
\write13<2598 2355 C>
\write13<5097 2841 C>
\write13<2544 2344 C>
\write13<5141 1936 C>
\write13<2445 2328 C>
\write13<5200 2872 C>
\write13<2309 2291 C>
\write13<5378 2935 C>
\write13<2131 1989 C>
\write13<5747 3099 C>
\write13<1866 2106 C>
\write13<6172 3684 C>
\write13<3820 1376 C>
\write13<3819 3722 C>
\write13<3825 1457 C>
\write13<3807 1195 C>
\write13<3838 3799 C>
\write13<3774 758 C>
\write13<3887 3964 C>
\write13<4059 4347 C>
\write13<4690 1477 C>
\write13<5041 2826 C>
\write13<2600 2355 C>
\write13<5090 2839 C>
\write13<2549 2342 C>
\write13<5181 2866 C>
\write13<2454 2320 C>
\write13<5334 2916 C>
\write13<2292 2285 C>
\write13<2090 2175 C>
\write13<5603 3040 C>
\write13<1843 3022 C>
\write13<6103 3337 C>
\write13<1435 1968 C>
\write13<3821 2590 C>
\write13<3819 3724 C>
\write13<3823 1454 C>
\write13<3808 1361 C>
\write13<3834 3810 C>
\write13<3874 4001 C>
\write13<3762 1144 C>
\write13<3991 4496 C>
\write13<3571 520 C>
\write13<4690 1477 C>
\write13<5042 2826 C>
\write13<2598 2354 C>
\write13<5095 2841 C>
\write13<2542 2340 C>
\write13<5196 2871 C>
\write13<2432 2314 C>
\write13<5368 2930 C>
\write13<2245 2275 C>
\write13<2038 1773 C>
\write13<5703 3087 C>
\write13<1843 3022 C>
\write13<1828 1965 C>
\write13<6280 3564 C>
\write13<3821 3722 C>
\write13<3824 1449 C>
\write13<3806 1351 C>
\write13<3847 3798 C>
\write13<3877 1526 C>
\write13<3754 1146 C>
\write13<3927 3958 C>
\write13<3577 658 C>
\write13<4193 4221 C>
\write13<5041 2826 C>
\write13<2600 2354 C>
\write13<5090 2842 C>
\write13<2550 2342 C>
\write13<5183 2872 C>
\write13<2456 2318 C>
\write13<5338 2930 C>
\write13<2299 2283 C>
\write13<2111 2179 C>
\write13<5548 3089 C>
\write13<5964 3401 C>
\write13<1347 2048 C>
\write13<3821 2590 C>
\write13<3822 1444 C>
\write13<3809 3762 C>
\write13<3800 3716 C>
\write13<3793 1328 C>
\write13<3874 3890 C>
\write13<3877 1526 C>
\write13<3704 1071 C>
\write13<4104 4125 C>
\write13<4850 4515 C>
\write13<5042 2827 C>
\write13<2598 2353 C>
\write13<5095 2843 C>
\write13<2543 2334 C>
\write13<2466 2110 C>
\write13<5195 2876 C>
\write13<2431 2296 C>
\write13<5366 2936 C>
\write13<2235 2248 C>
\write13<5659 3137 C>
\write13<6054 3757 C>
\write13<1502 2009 C>
\write13<3821 3722 C>
\write13<3818 1457 C>
\write13<3848 3799 C>
\write13<3783 1371 C>
\write13<3929 3963 C>
\write13<3693 1162 C>
\write13<3453 569 C>
\write13<4617 4242 C>
\write13<2600 2356 C>
\write13<5043 2824 C>
\write13<2551 2351 C>
\write13<5101 2833 C>
\write13<5145 2715 C>
\write13<2467 2345 C>
\write13<5219 2856 C>
\write13<2352 2321 C>
\write13<5431 2906 C>
\write13<2098 2242 C>
\write13<5758 3007 C>
\write13<1795 2218 C>
\write13<6731 3471 C>
\write13<892 1754 C>
\write13<3821 2590 C>
\write13<3823 3723 C>
\write13<3816 1457 C>
\write13<3859 3805 C>
\write13<3775 1372 C>
\write13<3964 3973 C>
\write13<3674 1184 C>
\write13<3452 782 C>
\write13<4586 4224 C>
\write13<2598 2356 C>
\write13<5045 2824 C>
\write13<2544 2351 C>
\write13<5111 2834 C>
\write13<5145 2715 C>
\write13<2453 2346 C>
\write13<5249 2860 C>
\write13<2318 2317 C>
\write13<5505 2914 C>
\write13<1988 2274 C>
\write13<5966 3091 C>
\write13<1555 2136 C>
\write13<3821 3723 C>
\write13<3822 1456 C>
\write13<3805 1371 C>
\write13<3844 3807 C>
\write13<3762 1179 C>
\write13<3908 3988 C>
\write13<3645 672 C>
\write13<4013 2935 C>
\write13<4139 4430 C>
\write13<4852 2523 C>
\write13<2735 1916 C>
\write13<2599 2353 C>
\write13<5043 2827 C>
\write13<2547 2337 C>
\write13<5101 2843 C>
\write13<2447 2303 C>
\write13<5210 2879 C>
\write13<2265 2239 C>
\write13<5410 2956 C>
\write13<1982 2080 C>
\write13<5710 3083 C>
\write13<6312 3768 C>
\write13<1275 1676 C>
\write13<3821 2590 C>
\write13<3821 1149 C>
\write13<3819 1359 C>
\write13<3824 3727 C>
\write13<3824 1454 C>
\write13<3844 669 C>
\write13<3863 3824 C>
\write13<3983 4051 C>
\write13<4013 2935 C>
\write13<4760 4764 C>
\write13<4852 2523 C>
\write13<2735 1916 C>
\write13<4936 3473 C>
\write13<2598 2353 C>
\write13<5046 2828 C>
\write13<2540 2334 C>
\write13<5112 2848 C>
\write13<2428 2297 C>
\write13<5243 2894 C>
\write13<2220 2227 C>
\write13<5528 2989 C>
\write13<1813 2001 C>
\write13<5977 3425 C>
\write13<895 851 C>
\write13<3821 1459 C>
\write13<3818 3722 C>
\write13<3829 3799 C>
\write13<3803 1382 C>
\write13<3859 3965 C>
\write13<3754 1219 C>
\write13<3953 4344 C>
\write13<3588 833 C>
\write13<5041 2825 C>
\write13<2600 2354 C>
\write13<5090 2834 C>
\write13<2550 2341 C>
\write13<2500 3244 C>
\write13<5179 2849 C>
\write13<2456 2315 C>
\write13<5302 2875 C>
\write13<2296 2264 C>
\write13<5491 2987 C>
\write13<1988 2138 C>
\write13<5907 3061 C>
\write13<1477 1724 C>
\write13<817 711 C>
\write13<3821 2590 C>
\write13<3820 3724 C>
\write13<3822 1456 C>
\write13<3807 1370 C>
\write13<3840 3811 C>
\write13<3768 1178 C>
\write13<3899 4002 C>
\write13<3670 657 C>
\write13<4124 4475 C>
\write13<2599 2354 C>
\write13<5043 2825 C>
\write13<2544 2339 C>
\write13<5097 2836 C>
\write13<2500 3244 C>
\write13<5196 2852 C>
\write13<2441 2308 C>
\write13<5332 2889 C>
\write13<2263 2245 C>
\write13<5510 3191 C>
\write13<1894 2081 C>
\write13<5775 3074 C>
\write13<1469 1496 C>
\write13
\write13<6486 4486 M>
\write13<(Set 2) Rshow>
\write13<6654 4486 B>
\write13<3816 3700 B>
\write13<3827 1479 B>
\write13<2615 2358 B>
\write13<5026 2822 B>
\write13<3815 3700 B>
\write13<2616 2359 B>
\write13<5027 2822 B>
\write13<2615 2358 B>
\write13<5026 2822 B>
\write13<3826 1480 B>
\write13<5026 2822 B>
\write13<2615 2358 B>
\write13<3825 1480 B>
\write13<3815 3701 B>
\write13<5026 2822 B>
\write13<2615 2358 B>
\write13<3825 1480 B>
\write13<3814 3701 B>
\write13<5026 2822 B>
\write13<2615 2358 B>
\write13<3826 1480 B>
\write13<5025 2821 B>
\write13<2614 2358 B>
\write13<5026 2822 B>
\write13<2615 2358 B>
\write13<3815 3700 B>
\write13<2615 2358 B>
\write13<5026 2822 B>
\write13<3816 3700 B>
\write13<3826 1479 B>
\write13<2615 2358 B>
\write13<5026 2822 B>
\write13
\write13
\write13
\write13
\write13<%%Trailer>
\write13<%%Pages: 1>
\closeout13
\openout14=4.ps
\write14<%!PS-Adobe-2.0>
\write14<%%Creator: gnuplot>
\write14<%%DocumentFonts: Helvetica>
\write14<%%BoundingBox: 50 50 554 770>
\write14<%%Pages: (atend)>
\write14<%%EndComments>
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14< 0 vshift R show } def>
\write14
\write14< dup stringwidth pop neg vshift R show } def>
\write14
\write14< dup stringwidth pop -2 div vshift R show } def>
\write14
\write14< {pop pop pop Solid {pop []} if 0 setdash} ifelse } def>
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14
\write14< currentlinewidth 2 div sub M>
\write14< 0 currentlinewidth V stroke } def>
\write14
\write14< hpt neg vpt neg V hpt vpt neg V>
\write14< hpt vpt V hpt neg vpt V closepath stroke>
\write14< P } def>
\write14
\write14< currentpoint stroke M>
\write14< hpt neg vpt neg R hpt2 0 V stroke>
\write14< } def>
\write14
\write14< 0 vpt2 neg V hpt2 0 V 0 vpt2 V>
\write14< hpt2 neg 0 V closepath stroke>
\write14< P } def>
\write14
\write14< hpt2 vpt2 neg V currentpoint stroke M>
\write14< hpt2 neg 0 R hpt2 vpt2 V stroke } def>
\write14
\write14< hpt neg vpt -1.62 mul V>
\write14< hpt 2 mul 0 V>
\write14< hpt neg vpt 1.62 mul V closepath stroke>
\write14< P } def>
\write14
\write14
\write14<%%EndProlog>
\write14<%%Page: 1 1>
\write14
\write14
\write14<50 50 translate>
\write14<0.100 0.100 scale>
\write14<90 rotate>
\write14<0 -5040 translate>
\write14<0 setgray>
\write14
\write14
\write14
\write14<840 2590 M>
\write14<6129 0 V>
\write14<3905 351 M>
\write14<0 4478 V>
\write14
\write14<840 351 M>
\write14<63 0 V>
\write14<6066 0 R>
\write14<-63 0 V>
\write14<756 351 M>
\write14<(-10) Rshow>
\write14<840 1471 M>
\write14<63 0 V>
\write14<6066 0 R>
\write14<-63 0 V>
\write14<-6150 0 R>
\write14<(-5) Rshow>
\write14<840 2590 M>
\write14<63 0 V>
\write14<6066 0 R>
\write14<-63 0 V>
\write14<-6150 0 R>
\write14<(0) Rshow>
\write14<840 3710 M>
\write14<63 0 V>
\write14<6066 0 R>
\write14<-63 0 V>
\write14<-6150 0 R>
\write14<(5) Rshow>
\write14<840 4829 M>
\write14<63 0 V>
\write14<6066 0 R>
\write14<-63 0 V>
\write14<-6150 0 R>
\write14<(10) Rshow>
\write14<840 351 M>
\write14<0 63 V>
\write14<0 4415 R>
\write14<0 -63 V>
\write14<840 211 M>
\write14<(-10) Cshow>
\write14<2372 351 M>
\write14<0 63 V>
\write14<0 4415 R>
\write14<0 -63 V>
\write14<0 -4555 R>
\write14<(-5) Cshow>
\write14<3905 351 M>
\write14<0 63 V>
\write14<0 4415 R>
\write14<0 -63 V>
\write14<0 -4555 R>
\write14<(0) Cshow>
\write14<5437 351 M>
\write14<0 63 V>
\write14<0 4415 R>
\write14<0 -63 V>
\write14<0 -4555 R>
\write14<(5) Cshow>
\write14<6969 351 M>
\write14<0 63 V>
\write14<0 4415 R>
\write14<0 -63 V>
\write14<0 -4555 R>
\write14<(10) Cshow>
\write14<840 351 M>
\write14<6129 0 V>
\write14<0 4478 V>
\write14<-6129 0 V>
\write14<840 351 L>
\write14<140 2590 M>
\write14
\write14<(Im) Cshow>
\write14
\write14<3904 71 M>
\write14<(Real) Cshow>
\write14<3904 4969 M>
\write14<(Figure 4) Cshow>
\write14
\write14<6486 4626 M>
\write14<(Set 1) Rshow>
\write14<6654 4626 C>
\write14<3905 3870 C>
\write14<3905 2011 C>
\write14<4212 2590 C>
\write14<3597 2590 C>
\write14<3593 2590 C>
\write14<4216 2590 C>
\write14<4222 2590 C>
\write14<3587 2590 C>
\write14<3577 2590 C>
\write14<4232 2590 C>
\write14<4244 2591 C>
\write14<3565 2591 C>
\write14<3548 2591 C>
\write14<4261 2591 C>
\write14<4282 2592 C>
\write14<3527 2592 C>
\write14<3499 2593 C>
\write14<4310 2593 C>
\write14<4346 2595 C>
\write14<3463 2595 C>
\write14<3416 2597 C>
\write14<4393 2597 C>
\write14<4458 2601 C>
\write14<3351 2601 C>
\write14<3261 2608 C>
\write14<4548 2608 C>
\write14<4682 2623 C>
\write14<3127 2623 C>
\write14<2910 2656 C>
\write14<4899 2656 C>
\write14<2501 2757 C>
\write14<5308 2757 C>
\write14<1507 3275 C>
\write14<6302 3275 C>
\write14<3905 3858 C>
\write14<3905 2011 C>
\write14<3597 2590 C>
\write14<4212 2590 C>
\write14<4216 2590 C>
\write14<3593 2590 C>
\write14<3587 2590 C>
\write14<4222 2590 C>
\write14<4231 2590 C>
\write14<3578 2590 C>
\write14<3565 2591 C>
\write14<4244 2591 C>
\write14<4260 2591 C>
\write14<3549 2591 C>
\write14<3528 2592 C>
\write14<4281 2592 C>
\write14<4308 2593 C>
\write14<3501 2593 C>
\write14<3466 2594 C>
\write14<4343 2594 C>
\write14<4389 2597 C>
\write14<3420 2597 C>
\write14<3359 2601 C>
\write14<4450 2601 C>
\write14<4536 2608 C>
\write14<3273 2608 C>
\write14<3148 2621 C>
\write14<4661 2621 C>
\write14<2948 2651 C>
\write14<4861 2651 C>
\write14<5219 2735 C>
\write14<2590 2735 C>
\write14<1786 3103 C>
\write14<6023 3103 C>
\write14
\write14
\write14
\write14
\write14<%%Trailer>
\write14<%%Pages: 1>
\closeout14
\end