% m211oh91.tex (prev.: *oh87.*) - Powers of Complex Numbers
%
\newcommand{\titlesize}{30pt} % for PcTeX fonts -usual: 30pt 
\input{m211oh.tex}
%
\vspace*{-20pt}
\begin{center}\tcr{\rmH Powers of Complex Numbers} \end{center} %\vspace{0pt}
%\LARGE           
\huge

\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
\newcommand{\vv}[1]{\vec{v}_{\scr #1}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\mult}{\mbox{mult}}
\newcommand{\be}{\beta}

\begin{description}  
\item[\tcr{Definition:}] If \mc{c_n = a_n + ib_n} is a sequence of \tcg{complex numbers} such that the limits
$$
\tcb{a = \lim_{n\ra\infty} a_n\quad\mbox{and}\quad
 b = \lim_{n\ra\infty} b_n}
$$
exist, then we say that \mc{c := a+ib} is the \tcr{\em limit}\/ of \mc{c_n} and write 
$$
\tcb{\lim_{n\ra\infty} c_n = a + ib.}
$$
\item[\tcr{Theorem 2:}] If \mc{\alpha \in \C} then

\tcr{\bf a)} If \mc{|\alpha| < 1} then \mc{\displaystyle\lim_{n\ra\infty} \alpha^n = 0.}

\tcr{\bf b)} If \mc{|\alpha| > 1} then \mc{\displaystyle\lim_{n\ra\infty} \alpha^n = \infty.}

\tcr{\bf c)} If \mc{|\alpha| = 1} then \mc{\displaystyle\lim_{n\ra\infty} \alpha^n} \tcg{exists} if and only if \mc{\alpha = 1}.

\item[\tcr{Corollary:}] \mc{\displaystyle\lim_{n\ra\infty} \alpha^n} \tcg{exists} if and only if \mc{\alpha = 1} or \mc{|\alpha| < 1}.

\item[\tcr{Note:}] \tcr{\bf a)} If \mc{|\alpha| <1} then the powers \mc{\alpha^n} \tcg{spiral in} to \mc{0}. 

\tcr{\bf b)} If \mc{|\alpha| > 1} then the powers \mc{\alpha^n} \tcg{spiral out} to \mc{\infty}.

\tcr{\bf c)} If \mc{|\alpha| = 1} and \mc{\alpha \ne 1} then the powers \mc{\alpha^n} \tcg{run around} on the \tcb{unit circle}. 
\end{description} 
\newpage

\begin{description}
\item[\tcr{\bf Example:}]
Powers of the numbers 
\end{description}
\LARGE
$$
\textstyle
\tcr{\al = \frac{5}{12}(2+i)}, \;
\tcb{\be = \frac{11}{24}(2+i)}\, \mbox{ and }
\tcg{\gamma = \cos(\frac{2\pi}{100})+i\sin(\frac{2\pi}{100}):}
$$ 
%\end{description}
\vspace{50pt}

\begin{center}
\normalsize
%\centering %\vspace{-1cm}
\setlength{\unitlength}{4.5cm}
\newcommand{\ci}{{\circle*{.025}}}
\newcommand{\cii}{{\circle{.025}}}
\newcommand{\ciii}{{\circle{.015}}}
\begin{picture}(3.0,3.0)(-1.5,-1.5)
% axes
%%\put(0,0){\circle*{.025}}
\put(-1.5,0){\line(1,0){3.0}}
\put(0,-1.5){\line(0,1){3.0}}
\multiput(-1.5,0)(.5,0){7}{\line(0,1){.02}}
\multiput(-.02,-1.5)(0,.5){7}{\line(1,0){.05}}
\put(-1.5,0){\makebox(0,0)[t]{\small$-1.5$\rule{0pt}{11pt}}}
\put(-1.0,0){\makebox(0,0)[t]{\small$-1$\rule{0pt}{11pt}}}
\put(-.5,0){\makebox(0,0)[t]{\small$-.5$\rule{0pt}{11pt}}}
\put(.5,0){\makebox(0,0)[t]{\small$.5$\rule{0pt}{11pt}}}
\put(1.,0){\makebox(0,0)[t]{\small$1$\rule{0pt}{11pt}}}
\put(1.5,0){\makebox(0,0)[t]{\small$1.5$\rule{0pt}{11pt}}}
\put(0,-1.5){\makebox(0,0)[l]{\small\hspace*{3pt}$-1.5$}}
\put(0,-1.0){\makebox(0,0)[l]{\small\hspace*{4pt}$-1$}}
\put(0,-.5){\makebox(0,0)[l]{\small\hspace*{4pt}$-.5$}}
\put(0,.5){\makebox(0,0)[l]{\small\hspace*{5pt}$.5$}}
\put(0,1.){\makebox(0,0)[l]{\small\hspace*{5pt}$1$}}
\put(0,1.5){\makebox(0,0)[l]{\small\hspace*{5pt}$1.5$}}
%
\tcr{
% inward spiral    %$z:= (5/12)*(2 + I)$, |z| =   .931694990
\put(.8333333333, .4166666667){\makebox(0,0)[r]{$\al$\hspace*{5pt}}}
\put(.8333333333, .4166666667)\ci
\put(.5208333333, .6944444444)\ci
\put(.5208333333, .6944444444){\makebox(0,0)[r]{$\al^2$\hspace*{5pt}}}
\put(.1446759259, .7957175926)\ci
\put( -.2109857253, .7233796296)\ci     
\put( -.4772296168, .5149056391)\ci    
\put( -.6122353636, .2302423590)\ci    
\put( -.6061304526,   -.06322943570)\ci    
\put( -.4787631123,   -.3052455517)\ci    
\put( -.2717836137,   -.4538559232)\ci    
\put( -.03737971010,   -.4914564417)\ci   %10  
\put( -.03737971010,   -.4914564417){\makebox(0,0)[tr]{$\al^{10}$}}
\put( .1736237589,   -.4251219139)\ci    
\put(  .3218205966,   -.2819250287)\ci    
\put( .3856525925,   -.1008456087)\ci    
\put( .3633961640,   .07665057296)\ci    
\put( .2708923979,   .2152905458)\ci    
\put( .1360392709,   .2922806206)\ci   
\put( -.008417532878,   .3002502134)\ci    
\put( -.1321188663,   .2467012058)\ci    
\put( -.2128912243,   .1505348105)\ci    
\put( -.2401321913,   .03674099864)\ci  
\put( -.2401321913,   .03674099864){\makebox(0,0)[br]{\small$\al^{20}$}}   
\put( -.2154189089,   -.06943758086)\ci    
\put( -.1505834320,   -.1476225294)\ci    
\put( -.06397680611,   -.1857618712)\ci    
\put( .02408677458,   -.1814585619)\ci    
\put( .09568004626,   -.1411793122)\ci    
\put( .1385580853,   -.07778274086)\ci    
\put( .1478745464,   -.007086415179)\ci    
\put( .1261814617,   .05570904836)\ci    
\put( .08193911459,   .09899981600)\ci    
\put( .02703267215,  .1166411444)\ci
\put( .02703267215, .1166411444) {\makebox(0,0)[lb]{\raisebox{3pt}{\scriptsize$\al^{30}$}}}        
\put( -.02607325005,  .1084645671)\ci    
\put( -.06692127799,  .07952328504)\ci    
\put( -.08890243376,  .03838553838)\ci    
\put( -.09007933579,  -.005054732086)\ci    
\put( -.07295997479,  -.04174533332)\ci 
%\put( -.07295997479,  -.04174533332){\makebox(0,0)[t]{$\al^{35}$}}      
\put( -.04340609011,  -.06518776726)\ci    
\put( -.009010172065,   -.07240901026)\ci    
\put( .02266194422,   -.06409508024)\ci    
\put( .04559123695,   -.04397009011)\ci    
\put( .05631356834,   -.01764539303)\ci    
\put( .05428022071,   .008759492617)\ci    
\put( .04158372867,   .02991633581)\ci    
\put( .02218796730,   .04225683345)\ci    
\put( .0008829588135,   .04445901426)\ci    
\put( -.01778879026,   .03741707805)\ci    
\put( -.03041444107,   .02376890243)\ci    
\put( -.03524907691,   .007134734915)\ci    
\put( -.03234703697,   -.008741502950)\ci    
\put( -.02331357125,   -.02076251786)\ci    
\put( -.01077692693,   -.02701608624)\ci}
%
% outward spiral:  z2:=(11/24)*(2 + I)    |z2| = 1.024864490
\tcb{
\put( .9166666667,   .4583333333){\makebox(0,0)[l]{\hspace*{5pt}$\beta$}}
\put( .9166666667,   .4583333333)\cii   
\put( .6302083333,   .8402777778)\cii
\put( .6302083333,   .8402777778){\makebox(0,0)[l]{\hspace*{5pt}$\beta^2$}}   
\put( .1925636574,   1.059100116)\cii   
\put( -.3089042004,   1.059100116)\cii   
\put( -.7685830701,   .8292606809)\cii   
\put( -1.084612293,   .4078883837)\cii   
\put( -1.181176778,   -.1232162826)\cii   
\put( -1.026271250,   -.6543209488)\cii   
\put( -.6408515444,   -1.070168526)\cii   
\put( -.09695334124,   -1.274711440)\cii
\put( -.09695334124,   -1.274711440){\makebox(0,0)[r]{$\beta^{10}\hspace*{3pt}$}}   
\put( .4953688472,   -1.212922435)\cii   
\put( 1.010010893,   -.8848015103)\cii   
\put( 1.331377344,   -.3481463920)\cii   
\put( 1.379996328,   .2910804232)\cii   
\put( 1.131584773,   .8993220383)\cii   
\put( .6250967747,   1.343021556)\cii   
\put( -.04254616980,   1.517605782)\cii   
\put( -.7345699723,   1.371638305)\cii   
\put( -1.302023365,   .9206572093)\cii   
\put( -1.615489305,   .2471750665)\cii
\put( -1.615489305,   .2471750665){\makebox(0,0)[l]{\hspace*{3pt}$\beta^{20}$}}}  
%
%  circle:
\tcg{
\put(1.,   0)\ciii  
\put(.9980267284,   .06279051954){\makebox(0,0)[l]{\small\hspace*{5pt}$\gamma$}}
\put(.9980267284,   .06279051954)\ciii   
\put(.9921147013,   .1253332336)\ciii
\put(.9921147013,   .1253332336){\makebox(0,0)[lb]{\small\hspace*{5pt}$\gamma^2$}}   
\put(.9822872507,   .1873813146)\ciii   
\put(.9685831611,   .2486898872)\ciii   
\put(.9510565160,   .3090169945)\ciii   
\put(.9297764859,   .3681245527)\ciii   
\put(.9048270524,   .4257792917)\ciii   
\put(.8763066800,   .4817536741)\ciii   
\put(.8443279255,   .5358267950)\ciii   
\put(.8090169945,   .5877852520)\ciii   
\put(.7705132427,   .6374239898)\ciii   
\put(.7289686273,   .6845471060)\ciii   
\put(.6845471059,   .7289686275)\ciii   
\put(.6374239897,   .7705132428)\ciii   
\put(.5877852520,   .8090169945)\ciii   
\put(.5358267951,   .8443279254)\ciii   
\put(.4817536743,   .8763066799)\ciii   
\put(.4257792918,   .9048270523)\ciii   
\put(.3681245521,   .9297764861)\ciii   
\put(.3090169945,   .9510565160)\ciii
\put(.3090169945,   .9510565160)
{\makebox(0,0)[lb]{\hspace*{7pt}$\gamma^{20}$}}     
\put(.2486898867,   .9685831613)\ciii   
\put(.1873813142,   .9822872508)\ciii   
\put(.1253332332,   .9921147014)\ciii   
\put(.06279051925,   .9980267284)\ciii   
\put(0,   1.)\ciii       
\put(-.06279051925,   .9980267284)\ciii   
\put(-.1253332332,   .9921147014)\ciii   
\put(-.1873813142,   .9822872508)\ciii   
\put(-.2486898867,   .9685831613)\ciii   
\put(-.3090169945,   .9510565160)\ciii   
\put(-.3681245521,   .9297764861)\ciii   
\put(-.4257792918,   .9048270523)\ciii   
\put(-.4817536743,   .8763066799)\ciii   
\put(-.5358267951,   .8443279254)\ciii   
\put(-.5877852520,   .8090169945)\ciii   
\put(-.6374239897,   .7705132428)\ciii   
\put(-.6845471059,   .7289686275)\ciii   
\put(-.7289686273,   .6845471060)\ciii   
\put(-.7705132427,   .6374239898)\ciii   
\put(-.8090169945,   .5877852520)\ciii   
\put(-.8443279255,   .5358267950)\ciii   
\put(-.8763066800,   .4817536741)\ciii   
\put(-.9048270524,   .4257792917)\ciii   
\put(-.9297764859,   .3681245527)\ciii   
\put(-.9510565160,   .3090169945)\ciii   
\put(-.9685831611,   .2486898872)\ciii   
\put(-.9822872507,   .1873813146)\ciii   
\put(-.9921147013,   .1253332336)\ciii   
\put(-.9980267284,   .06279051954)\ciii   
\put(-1.,   0)\ciii        
\put(-.9980267284,   -.06279051954)\ciii   
\put(-.9921147013,   -.1253332336)\ciii   
\put(-.9822872507,   -.1873813146)\ciii   
\put(-.9685831611,   -.2486898872)\ciii   
\put(-.9510565160,   -.3090169945)\ciii   
\put(-.9297764859,   -.3681245527)\ciii   
\put(-.9048270524,   -.4257792917)\ciii   
\put(-.8763066800,   -.4817536741)\ciii   
\put(-.8443279255,   -.5358267950)\ciii   
\put(-.8090169945,   -.5877852520)\ciii   
\put(-.7705132427,   -.6374239898)\ciii   
\put(-.7289686273,   -.6845471060)\ciii   
\put(-.6845471059,   -.7289686275)\ciii   
\put(-.6374239897,   -.7705132428)\ciii   
\put(-.5877852520,   -.8090169945)\ciii   
\put(-.5358267951,   -.8443279254)\ciii   
\put(-.4817536743,   -.8763066799)\ciii   
\put(-.4257792918,   -.9048270523)\ciii   
\put(-.3681245521,   -.9297764861)\ciii   
\put(-.3090169945,   -.9510565160)\ciii   
\put(-.2486898867,   -.9685831613)\ciii   
\put(-.1873813142,   -.9822872508)\ciii   
\put(-.1253332332,   -.9921147014)\ciii   
\put(-.06279051925,   -.9980267284)\ciii   
\put(0,   -1.)\ciii   
\put(.06279051925,   -.9980267284)\ciii   
\put(.1253332332,   -.9921147014)\ciii   
\put(.1873813142,   -.9822872508)\ciii   
\put(.2486898867,   -.9685831613)\ciii   
\put(.3090169945,   -.9510565160)\ciii   
\put(.3681245521,   -.9297764861)\ciii   
\put(.4257792918,   -.9048270523)\ciii   
\put(.4817536743,   -.8763066799)\ciii   
\put(.5358267951,   -.8443279254)\ciii   
\put(.5877852520,   -.8090169945)\ciii   
\put(.6374239897,   -.7705132428)\ciii   
\put(.6845471059,   -.7289686275)\ciii   
\put(.7289686273,   -.6845471060)\ciii   
\put(.7705132427,   -.6374239898)\ciii 
\put(.8090169945,   -.5877852520)\ciii 
\put(.8090169945,   -.5877852520){\makebox(0,0)[l]{\small\hspace*{3pt}$\gamma^{90}$}}  
\put(.8443279255,   -.5358267950)\ciii    
\put(.8763066800,   -.4817536741)\ciii   
\put(.9048270524,   -.4257792917)\ciii   
\put(.9297764859,   -.3681245527)\ciii   
\put(.9510565160,   -.3090169945)\ciii   
\put(.9685831611,   -.2486898872)\ciii   
\put(.9822872507,   -.1873813146)\ciii   
\put(.9921147013,   -.1253332336)\ciii   
\put(.9980267284,   -.06279051954)\ciii   
\put(1., 0)\ciii}
\end{picture}
\iffalse
\caption{Powers of the numbers $\al = \frac{5}{12}(2+i)$, $\be = \frac{11}{24}(2+i)$ and
$\gamma = \cos(\frac{2\pi}{100})+i\sin(\frac{2\pi}{100})$} 
\label{F7.21}
\end{figure}
\fi
\end{center}
\end{document}                          