
\magnification=1100
\overfullrule0pt

\input amssym.def
\input prepictex
\input pictex
\input postpictex


% ********************* Definitions ************************************

%\def\widetilde{\mathaccent"0365 }

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\resultno=0
\bigskip\noindent{\sectionfont #1.  #2}~\medbreak}

\def\subsection #1\par{\bigskip\noindent{\it  #1} \medbreak}

%******************* MATHEMATICAL LABELS **************************

\def\prop{ \global\advance\resultno by 1
\medskip\noindent{\bf Proposition \the\sectno.\the\resultno. }\sl}
\def\lemma{ \global\advance\resultno by 1
\medskip\noindent{\bf Lemma \the\sectno.\the\resultno. }
\sl}
\def\fact{ \global\advance\resultno by 1
\medskip\noindent{\bf Fact \the\sectno.\the\resultno. }
\sl}

\def\remark{ \global\advance\resultno by 1
\medskip\noindent{\bf Remark \the\sectno.\the\resultno. }}
\def\example{ \global\advance\resultno by 1
\medskip\noindent{\bf Example \the\sectno.\the\resultno. }\sl}
\def\cor{ \global\advance\resultno by 1
\medskip\noindent{\bf Corollary \the\sectno.\the\resultno. }\sl}
\def\thm{ \global\advance\resultno by 1
\medskip\noindent{\bf Theorem \the\sectno.\the\resultno. }\sl}
\def\defn{ \global\advance\resultno by 1
\medskip\noindent{\it Definition \the\sectno.\the\resultno. }\slrm}
\def\endthm{\rm\medskip}
\def\thmend{\rm\medskip}
\def\endlemma{\rm\medskip}
\def\endfact{\rm\medskip}
\def\endexample{\rm\medskip}
\def\endprop{\rm\medskip}
\def\endcor{\rm\medskip}
\def\pf{\rm\smallskip\noindent{\it Proof. }}
\def\endpf{\qed\hfil\medskip}
\def\pfend{\qed\hfil\medskip}
\def\note{\smallbreak\noindent{Note:}}
\def\enddefn{\rm\medskip}

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%high strut:
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%low strut:
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\setbox\strutLbox=\hbox{\vrule height 1pt depth5pt width0pt}
\def\strutL{\relax\copy\strutLbox}



% hack to ignore lots of typed stuff....
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% ******************  QED SIGNS  *********************************

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\def\whiteslug{\bf $ \square $ \rm}  % open square


%*************** EQUATIONS WITH NUMBERS **************

\def\formula{\global\advance\resultno by 1
\eqno{(\the\sectno.\the\resultno)}}
\def\formulano{\global\advance\resultno by 1 (\the\sectno.\the\resultno)}
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\the\sectno.\the\resultno. }
\def\lformula{\global\advance\resultno by 1
\leqno(\the\sectno.\the\resultno)}

%************Commutative diagrams**********************

\def\mapright#1{\smash{\mathop
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\def\mapleftright#1{\smash{\mathop
        {\longleftrightarrow}\limits^{#1}}}


\def\mapsrightto#1{\smash{\mathop
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%********** DATING ******************************************
\def\monthname {\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}

\newcount\mins  \newcount\hours  \hours=\time \mins=\time
\def\now{\divide\hours by60 \multiply\hours by60 \advance\mins by-\hours
     \divide\hours by60         % NOTE: \divide only gives integer answers.
     \ifnum\hours>12 \advance\hours by-12
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ P.M.\else
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ A.M.\fi}
\def\today {\monthname\ \number\day, \number\year}


%**************** PAGE HEADERS *************************

\nopagenumbers
\def\runningtitle{\smallcaps genesis}
\headline={\ifnum\pageno>1\eoheadline\else\firstheadline\fi}
\def\names{\smallcaps a.\ ram}
%\def\firstheadline{\noindent Preliminary Draft \hfill  \today}
\def\firstheadline{}
\def\eoheadline{\ifodd\pageno\oddheadline\else\evenheadline\fi}
\def\oddheadline{\tenrm\hfil\runningtitle\hfil\folio}
\def\evenheadline{\tenrm\folio\hfil{\names}\hfil}


%**************** TITLE *************************
\vphantom{$ $}  %My kludge to get the first page to move down a bit
\vskip.75truein
\centerline{\titlefont An approach to ``early trascendentals''}
\bigskip
\centerline{\rm Arun Ram}
%${}^\ast$ 
\centerline{Department of Mathematics}
\centerline{University of Wisconsin, Madison}
\centerline{Madison, WI 53706 USA}
\centerline{{\tt ram@math.wisc.edu}}
\medskip
\centerline{Version: \today}

%\footnote{}{\tinyrm 
%${}^\ast$ 
%Research partially supported by the National Security Agency
%and by EPSRC Grant GR K99015 at the Newton Institute for
%Mathematical Sciences.}
\footnote{}{\tinyrm
\noindent {Keywords:} exponential functions, trigonometric functions}

\bigskip

%**************** ABSTRACT *************************
%\noindent{\bf Abstract.}


\bigskip\noindent
{\bf The function ${\rm god}(t)$}

\bigskip\noindent
There is one function that
\smallskip\noindent
\itemitem{(a)} in the Beginning, created something from nothing, and
\smallskip\noindent
\itemitem{(b)} is ``unchanging'', or rather, its change is itself.
\smallskip\noindent
Through the ages thinkers have contemplated this function
and nowadays it is common to write (a) and (b) in abbreviated form,
$$\hbox{(a$'$)}~~~~{\rm god}(0)=1,
\qquad\qquad\hbox{and}\qquad\qquad
\hbox{(b$'$)}~~~~{d~{\rm god}(t)\over dt} = {\rm god}(t),$$
but the meaning is still the same.

\smallskip
Two of the children of god are eve and adam:
$${\rm god}(it) = {\rm adam}(t) + i ~{\rm eve}(t).$$


\bigskip\noindent
{\bf Trying to understand ${\rm god}(t)$}

\bigskip
If we try to ``understand'' god in ``normal'' terms,
$${\rm god}(t) = a_0 + a_1t +a_2t^2+a_3t^3+ \cdots,$$
then
$$\matrix{
\hbox{since} &\qquad
&{\rm god}(0)=1,\hfill &\qquad &a_0=1,\qquad \hfill&\hbox{and}\hfill \cr
\cr
\cr
\hbox{since} &\qquad
&\displaystyle{d{\rm god}(t)\over dt} = {\rm god}(t), \hfill
&\qquad &a_1=a_0, \hfill &\hbox{and}\hfill\cr
&&&&2a_2 = a_1, \hfill &\hbox{and}\hfill\cr
&&&&3a_3 = a_2, \hfill &\hbox{and}\hfill\cr
&&&&4a_4 = a_3, \hfill &\hbox{and}\hfill\cr
&&&&5a_5 = a_4, \hfill &\hbox{$\ldots,$ etc.,}\hfill\cr
}$$
and so
$${\rm god}(t)= 
1~+~t~+~{1\over 2!}t^2~+~{1\over 3!}t^3~+~{1\over 4!}t^4 ~+~ \cdots,$$
which illustrates that 
${\rm god}(t)$ exists everywhere and goes on forever.

\bigskip\noindent
{\bf An amazing thing about ${\rm god}(t)$}

\bigskip
One of the amazing things about god is that
$${\rm god}(t+s) = {\rm god}(t)~{\rm god}(s).$$
To see why god is this way 
suppose that there is a ``different'' function such that
\smallskip\noindent
\itemitem{(a$''$)} is ``unchanging''
\qquad $\displaystyle{ \left( \hbox{i.e.} \quad
{d~\widetilde{{\rm god}}(t)\over dt} = \widetilde{{\rm god}}(t)\right) }$, 
\qquad and
\smallskip\noindent
\itemitem{(b$''$)} in the Beginning, was the way that god is after
$s$ millenia
\qquad $(~\hbox{i.e.}\quad \widetilde{{\rm god}}(0)={\rm god}(s)~~)$.
\smallskip\noindent
By the chain rule,
$${d~{\rm god}(t+s)\over dt} = {\rm god}(t+s)
\qquad\hbox{and}\qquad
{\rm god}(0+s) = {\rm god}(s),$$
and so
$${\rm god}(t+s) = \widetilde{{\rm god}}(t).$$
Also,
$${d~({\rm god}(t){\rm god}(s))\over dt}
= {\rm god}(t){\rm god}(s),
\qquad\hbox{and}\qquad
{\rm god}(0){\rm god}(s) = {\rm god}(s),$$
and so
$${\rm god}(t){\rm god}(s) = \widetilde{{\rm god}}(t)= {\rm god}(t+s).$$

\bigskip\noindent
{\bf What about ${\rm adam}(t)$ and ${\rm eve}(t)$?}

\bigskip
$$\matrix{
{\rm god}(it) 
&= &1 &+it &\displaystyle{
+{(it)^2\over 2!} }&\displaystyle{
+{(it)^3\over 3!} }&\displaystyle{
+{(it)^4\over 4!}}
&\displaystyle{
+{(it)^5\over 5!} }&+ &\cdots \cr
\cr
\cr
&= &1 &&\displaystyle{
+{i^2t^2\over 2!} }&&\displaystyle{
+{i^4t^4\over 4!} }&&\displaystyle{
+{i^6t^6\over 6!} }&&+\cdots \cr
&&&+it &&\displaystyle{
+{i^3t^3\over 3!} }&&\displaystyle{
+{i^5t^5\over 5!} }&&\displaystyle{
+{i^7t^7\over 7!} }
&&+\cdots \cr
\cr
\cr
&= &1 &&\displaystyle{
-{t^2\over 2!} }&&\displaystyle{
+{t^4\over 4!} }&&\displaystyle{
-{t^6\over 6!} }&&+\cdots \cr
&&&+it &&\displaystyle{
-{it^3\over 3!} }&&\displaystyle{
+{it^5\over 5!} }&&\displaystyle{
-{it^7\over 7!} }
&&+\cdots \cr
}$$
$$
= \left(1-{t^2\over 2!}+{t^4\over 4!}-{t^6\over 6!}+{t^8\over 8!}
-\cdots\right) 
+ i\left(t-{t^3\over 3!}+{t^5\over 5!}-{t^7\over 7!}+\cdots\right)
\hskip.5in 
$$
and, since adam and eve are the children of god,
$$\hbox{~~i.e.\qquad
because \quad${\rm god}(it) = {\rm adam}(t) + i~{\rm eve}(t)$~,\qquad}$$
we see that
$$\eqalign{
{\rm adam}(t)  
&= 1-{t^2\over 2!}+{t^4\over 4!}-{t^6\over 6!}+{t^8\over 8!}+\cdots, 
\qquad\qquad\qquad\hbox{and} \cr
\cr
{\rm eve}(t) 
&=t-{t^3\over 3!}+{t^5\over 5!}-{t^7\over 7!}+{t^9\over 9!}+\cdots, \cr
}$$
from which it follows that
$$\matrix{
{\rm adam}(0)=0, &\qquad\qquad &{\rm eve}(0)=1, \cr
\cr
{\rm adam}(-t)=-{\rm adam}(t), &\qquad\qquad 
&{\rm eve}(-t)={\rm eve}(t), \cr
\cr
\displaystyle{{d~{\rm adam}(t)\over dt}={\rm eve}(t)}, &\qquad\qquad 
&\displaystyle{{d~{\rm eve}(t)\over dt}=-{\rm adam}(t)}. \cr
}$$
So, adam and eve are complete opposites and identical twins at the
same time.

\bigskip\noindent
{\bf Complete opposites and 
identical twins at the same time, another manifestation}

\medskip
$$\eqalign{
1 &= {\rm god}(0) = {\rm god}(it-it)
={\rm god}(it+i(-t)) = {\rm god}(it){\rm god}(i(-t)) \cr
&= ({\rm adam}(t)+i~{\rm eve}(t))({\rm adam}(-t)+i~{\rm eve}(-t)) \cr
&= ({\rm adam}(t)+i~{\rm eve}(t))({\rm adam}(t)-i~{\rm eve}(t)) \cr
&= ({\rm adam}(t))^2+({\rm eve}(t))^2, \cr
\cr
\hbox{i.e.} \qquad\qquad
1 &= ({\rm adam}(t))^2+({\rm eve}(t))^2. \cr
}$$

\bigskip\noindent
{\bf Through the ages: where are we now?}

\bigskip\noindent
Let $x = {\rm eve}(t)$ and $y = {\rm adam}(t)$.
\medskip\noindent
\itemitem{(A)} In the Beginning the point $(x,y)$ was at $
({\rm adam}(0),{\rm eve}(0))=(1,0)$,\quad and
\medskip\noindent
since $1={\rm adam}(t))^2+({\rm eve}(t))^2$, \qquad
$x^2+y^2=1$,\qquad and 
\medskip\noindent
\itemitem{(B)} adam and eve travel through the ages on a circle of radius 1.
$$
\beginpicture
\setcoordinatesystem units <2cm,2cm>         % sets scale
\setplotarea x from -1.7 to 1.7, y from -1.7 to 1.7    % sets plot size up
\plot -1.5 0  1.5 0 /
\plot 0 -1.5  0 1.5 /
\put{$({\rm adam}(0),{\rm eve}(0))$}[bl] at 1.1 0.1      %
\put{$\bullet$} at 1 0      %
\put{$y$} at -.2 1.4      %
\put{$x$} at 1.4 -.2      %
%Circle
\ellipticalarc axes ratio 1:1 360 degrees from 1 0 center at 0 0
\endpicture
$$

\noindent
Where are they after $d$ millenia?
$$\eqalign{
\matrix{
\hbox{The distance traveled}\cr
\hbox{after $d$ millenia}\cr}
&= \int_{t=0}^{t=d} ds
= \int_{t=0}^{t=d} \displaystyle{
\sqrt{\left({dx\over dt}\right)^2
+\left({dy\over dt}\right)^2} ~dt } \cr
\cr
&= \int_{t=0}^{t=d} \displaystyle{
\sqrt{\left({d~{\rm adam}(t)\over dt}\right)^2
+\left({d~{\rm eve}(t)\over dt}\right)^2} ~dt } \cr
\cr
&= \int_{t=0}^{t=d} 
\sqrt{({\rm eve})^2+(-{\rm adam}(t))^2} ~dt \cr
\cr
&= \int_{t=0}^{t=d}\sqrt{1} ~dt
= \int_{t=0}^{t=d} dt = ~t~\Big\vert^{t=d}_{t=0}
=d-0=d, \cr
}$$
and so
$$
\matrix{
{\rm adam}(t) ~~~=
~~\hbox{$x$-coordinate of the point on a circle of radius 1}\hfill 
\cr
\phantom{{\rm adam}(t) ~~~=~~}
~~\hbox{which is distance $d$ from the point (1,0),}\hfill 
&\qquad\qquad\hbox{and} \cr
\cr
{\rm eve}(t) ~~~=
~~\hbox{$y$-coordinate of the point on a circle of radius 1} \hfill\cr
\phantom{{\rm adam}(t) ~~~=~~}
~~\hbox{which is distance $d$ from the point (1,0).} \hfill\cr
}$$
$$
\beginpicture
\setcoordinatesystem units <2cm,2cm>         % sets scale
\setplotarea x from -1.7 to 1.7, y from -1.7 to 1.7    % sets plot size up
\plot -1.5 0  1.5 0 /
\plot 0 -1.5  0 1.5 /
\plot 0 0  0.5 .866025 /
\plot 0.5 0  0.5 .866025 /
\put{$({\rm adam}(0),{\rm eve}(0))$}[bl] at 1.1 0.1      %
\put{$({\rm adam}(d),{\rm eve}(d))$}[bl] at .6 0.9      %
\put{$\bullet$} at .5 .866025 %
\put{$\bullet$} at 1 0      %
\put{$y$} at -.2 1.4      %
\put{$x$} at 1.4 -.2      %
%Circle
\ellipticalarc axes ratio 1:1 360 degrees from 1 0 center at 0 0
\endpicture
$$
The triangle in this picture is
$$
\beginpicture
\setcoordinatesystem units <2cm,2cm>         % sets scale
\setplotarea x from -0.3 to 1.2, y from -0.3 to 1.2    % sets plot size up
\plot 0 0  0.5 0 /
\plot 0 0  0.5 .866025 /
\plot 0.5 0  0.5 .866025 /
\put{$1$}[r] at 0.1 0.4      %
\put{${\rm adam}(d)$}[t] at 0.25 -0.1      %
\put{${\rm eve}(d)$}[l] at .6 0.4      %
\endpicture
\qquad\qquad\qquad\qquad
\beginpicture
\setcoordinatesystem units <2cm,2cm>         % sets scale
\setplotarea x from -0.3 to 1.2, y from -0.3 to 1.2    % sets plot size up
\plot 0 0  0.5 0 /
\plot 0 0  0.5 .866025 /
\plot 0.5 0  0.5 .866025 /
\put{hypotenuse}[r] at 0.1 0.4      %
\put{adjacent}[t] at 0.25 -0.1      %
\put{opposite}[l] at .6 0.4      %
\endpicture
$$
and so
$$
{\rm adam}(d) = {\hbox{opposite}\over \hbox{hypotenuse}}
\qquad\hbox{and}\qquad
{\rm eve}(d) = {\hbox{adjacent}\over \hbox{hypotenuse}}
$$
for a right triangle with angle $d$.

\bigskip\noindent
{\bf Some remarks on society}

\bigskip\noindent
{\bf 1.}  It is interesting to note that our school systems
like to introduce our children to ${\rm adam}(t)$
and ${\rm eve}(t)$ but prefer to hide from my child
how close they really are to ${\rm god}(t)$.

\bigskip\noindent
{\bf 2.}  Mathematicians are a cloistered group and prefer
to study god, adam, and eve in anonymity.  In the
mathematical literature
$$\matrix{
{\rm god}(t)\qquad\hfill
&\hbox{is usually called}\qquad \hfill
&e^t~, \hfill\cr
{\rm adam}(t)\hfill
&\hbox{is usually termed} \qquad
&\cos t~,\hfill 
&\qquad\hbox{and} \hfill \cr
{\rm eve}(t)\hfill
&\hbox{is usually called} \hfill
&\sin t~.  \hfill \cr
}$$

 
\vfill\eject
\end




% Last Edit 7 February 1997

%            This document is written in Plain TeX
%            The macros: prepictex.tex, pictex.tex, and postpictex.tex are also
%             required for the full compilation of the document.

\magnification=1000
\overfullrule0pt
\nopagenumbers

\input prePicTeX
\input PicTeX
\input postPicTeX
\input amssym.def

\vphantom{$ $}  %My kludge to get the first page to move down a bit
\vskip2truein


$$
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\plot -4.5 2.5 -4.5 1 /
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%\plot  3 2   3 .5 /
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% single crossing
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%0 crossing
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$$


$$
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\put{$\bullet$} at -3 2      %
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\put{$\bullet$} at  2 0.5          %
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%Flagpole
\plot -4.5 2.5 -4.5 1.62 /
\plot -4.5 1.38 -4.5 1.12 /
\plot -4.5 .88  -4.5 0 /
\plot -4.25 2.5 -4.25 1.62 /
\plot -4.25 1.38 -4.25 1.12 /
\plot -4.25 .88  -4.25 0 /
\ellipticalarc axes ratio 1:1 360 degrees from -4.5 2.5 center 
at -4.375 2.5
\put{$*$} at -4.375 2.5  
\ellipticalarc axes ratio 1:1 180 degrees from -4.5 0 center 
at -4.375 0 
% Vertical edges
%\plot -3 2  -3 .5 /
\plot -2 2  -2 .5 /
\plot -1 2  -1 .5 /
%\plot  0 2   0 .5 /
%\plot  1 2   1 .5 /
\plot  2 2   2 .5 /
%\plot  3 2   3 .5 /
\setquadratic
%0 crossing
\plot  3 .5  2.98 .7  2.8 .8   /  %bottom corner
\plot  3 2  2.98 1.8  2.8 1.7 /  %top corner
\setlinear
%top edge
\plot 2.8 1.7  2.15 1.7 /
\plot 1.85 1.7  1.15 1.7 /
\plot 0.85 1.7  0.15 1.7 /
\plot -0.15 1.7 -0.85 1.7 /
\plot -1.15 1.7 -1.85 1.7 /
\plot -2.15 1.7 -2.85 1.7 /
\plot -3.15 1.7  -4.1 1.7 /
\ellipticalarc axes ratio 2:1 180 degrees from -4.65 1.7  center 
at -4.65 1.6 
\plot -4.65 1.5  -3.8 1.5 /
%bottom edge
\plot 2.8 .8  2.15 .8 /
\plot 1.85 .8  1.15 .8 /
\plot 0.85 .8  0.15 .8 /
\plot -0.15 .8 -0.85 .8 /
\plot -1.15 .8 -1.85 .8 /
\plot -2.15 .8 -2.85 .8 /
\plot -3.15 .8  -4.1 .8 /
\ellipticalarc axes ratio 2:1 180 degrees from -4.65 1  center 
at -4.65 .9 
\plot -4.65 1  -3.8 1 /
\setquadratic
\plot  -3.8 1.5  -3.7 1.45 -3.5 1.25  -3.05 .8  -3 .5 /
\plot  -3.4 1.35  -3.05 1.7  -3 2 /
\plot  -3.8 1  -3.7 1.05  -3.6 1.15 /
% single crossing
\plot  0 .5  .05 .8  .4 1.15 /
\plot  .6 1.35  .95 1.7  1 2 /
\plot 0 2  .05 1.7  .5 1.25  .95 .8  1 .5 /
\endpicture
$$


$$
\beginpicture
\setcoordinatesystem units <1cm,1cm>         % sets scale
\setplotarea x from -5 to 5, y from -2 to 2    % sets plot size up
\put{$\bullet$} at -3 2      %
\put{$\bullet$} at -2 2      %
\put{$\bullet$} at -1 2      %
\put{$\bullet$} at  0 2      %   Top dots
\put{$\bullet$} at  1 2      %
\put{$\bullet$} at  2 2      %
\put{$\bullet$} at  3 2      %   
\put{$\bullet$} at -3 .5          %
\put{$\bullet$} at -2 .5          %
\put{$\bullet$} at -1 .5          %
\put{$\bullet$} at  0 .5          %  Bottom dots
\put{$\bullet$} at  1 .5          %
\put{$\bullet$} at  2 .5          %
\put{$\bullet$} at  3 .5          %
%Border of annulus
\plot -3 .5 3 .5 / 
\plot -3 -.5  3 -.5 / 
\ellipticalarc axes ratio 2:1 180 degrees from -3 .5 center at -3 0
\ellipticalarc axes ratio 2:1 180 degrees from 3 -.5 center at 3 0
\plot -3 2  3 2 /
\plot -3 -2  3 -2 /
\ellipticalarc axes ratio 5:4 180 degrees from -3 2 center at -3 0
\ellipticalarc axes ratio 5:4 180 degrees from 3 -2 center at 3 0
% Vertical edges
%\plot -3 2  -3 .5 /
\plot -2 2  -2 .5 /
\plot -1 2  -1 .5 /
%\plot  0 2   0 .5 /
%\plot  1 2   1 .5 /
\plot  2 2   2 .5 /
%\plot  3 2   3 .5 /
\setquadratic
%0 crossing
\plot  3 .5  3.02 .8  3.2 1   /  %bottom corner
\plot  3 2  3.05 1.7  3.2 1.5 /  %top corner
\plot  -3.8 1.5  -3.7 1.45 -3.5 1.25  -3.05 .8  -3 .5 /
\plot  -3.4 1.35  -3.05 1.7  -3 2 /
\plot  -3.8 1  -3.7 1.05  -3.6 1.15 /
\ellipticalarc axes ratio 1.8:2 180 degrees from -3.8 1 center at -3.8 0
\ellipticalarc axes ratio 2.8:2 180 degrees from 3.2 -1 center at 3.2 0
\ellipticalarc axes ratio 2.8:3 180 degrees from -3.8 1.5 center at -3.8 0
\ellipticalarc axes ratio 3.8:3 180 degrees from 3.2 -1.5 center at 3.2 0
\setlinear
\plot -3.8 -1  3.2 -1 /
\plot -3.8 -1.5  3.2 -1.5 /
\setquadratic
% single crossing
\plot  0 .5  .05 .8  .4 1.15 /
\plot  .6 1.35  .95 1.7  1 2 /
\plot 0 2  .05 1.7  .5 1.25  .95 .8  1 .5 /
\endpicture
$$
\bigskip
\centerline{\bf Figure 1.}



\vfill\eject
\end

