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% (defun u () (interactive) (find-latex-upload-links "2024-1-C2-numeros-complexos"))
% (defun v () (interactive) (find-2a '(e) '(d)))
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% (find-pdf-page   "~/LATEX/2024-1-C2-numeros-complexos.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-numeros-complexos.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-numeros-complexos.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-1-C2-numeros-complexos.pdf")
%   file:///home/edrx/LATEX/2024-1-C2-numeros-complexos.pdf
%               file:///tmp/2024-1-C2-numeros-complexos.pdf
%           file:///tmp/pen/2024-1-C2-numeros-complexos.pdf
%  http://anggtwu.net/LATEX/2024-1-C2-numeros-complexos.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2024-1-C2-numeros-complexos" "2" "c2m241nc" "c2nc")

% «.defs»		(to "defs")
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% «.defs-pict2e»	(to "defs-pict2e")
% «.defs-maxima»	(to "defs-maxima")
% «.defs-V»		(to "defs-V")
% «.title»		(to "title")
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% «.partes-de-cima»	(to "partes-de-cima")
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\documentclass[oneside,12pt]{article}
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\usepackage{amssymb}
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%
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\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
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\def\drafturl{http://anggtwu.net/2024.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

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% (find-LATEX "2024-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% «defs-T-and-B»  (to ".defs-T-and-B")
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\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
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%L dofile "Piecewise2.lua"           -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
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% «defs-maxima»  (to ".defs-maxima")
%L dofile "Maxima2.lua"              -- (find-angg "LUA/Maxima2.lua")
\pu

% «defs-V»  (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu


%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
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\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.1}

\bsk

Aula 31: revisão de números complexos

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m241ncp 2 "links")
% (c2m241nca   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{15cm}\firstcol{

% (find-books "__analysis/__analysis.el" "stewart-pt" "1020" "17.1 Equações Lineares de Segunda Ordem")
% (find-books "__analysis/__analysis.el" "stewart-pt" "1034" "subamortecimento")
% (find-books "__analysis/__analysis.el" "stewart-pt" "51" "H Números Complexos")
\par \Ca{StewPtCap17p6} (p.1020) Equações diferenciais de 2ª ordem
\par \Ca{StewPtCap17p20} (p.1034) Caso 3: subamortecimento
\par \Ca{StewPtApendiceHp5} (p.A51) Apêndice H: Números complexos

\ssk

% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "105" "3. Equações lineares de segunda")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "111" "operador diferencial")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "113" "princípio da superposição")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "121" "3.3. Raízes complexas")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "123" "Figura 3.3.1")
\par \Ca{BoyceDip3p5} (p.105) Capítulo 3: Equações lineares de 2ª ordem
\par \Ca{BoyceDip3p11} (p.111) Seção 3.2: o operador diferencial $L$
\par \Ca{BoyceDip3p13} (p.113) Teorema 3.2.2: o princípio da superposição
\par \Ca{BoyceDip3p21} (p.121) 3.3. Raízes complexas da equação característica
\par \Ca{BoyceDip3p23} (p.123) Figura 3.3.1

\ssk

% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "173" "4.3" "coeficientes constantes")
%\par \Ca{ZillCullenCap4p33} (p.173) 4.3. Equações lineares homogêneas com coeficientes constantes

% (find-books "__analysis/__analysis.el" "boyce-diprima" "103" "3 Second-Order Linear")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "110" "differential operator")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "112" "Theorem 3.2.2" "Superposition")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "120" "3.3 Complex Roots")
\par \Ca{BoyceDipEng3p4} (p.103) Chapter 3: Second-order linear ODEs
\par \Ca{BoyceDipEng3p11} (p.110) Section 3.2: the differential operator $L$
\par \Ca{BoyceDipEng3p13} (p.112) Theorem 3.2.2: principle of superposition
\par \Ca{BoyceDipEng3p21} (p.120) 3.3 Complex Roots of the Characteristic Equation
\par \Ca{BoyceDipEng3p24} (p.123) Figure 3.3.1


% (find-angg ".emacs" "c2q191" "20190524")
% (find-angg ".emacs" "c2q192" "60" "20190920")
% (find-c2q222page 45 "nov23: Números complexos")
% (find-c2q231page 50 "jun23: Oscilações")
% (c2q191 31 "20190524" "E = c + is")

% (find-SUBSfile "2021aulas-por-telegram.lua" "14:16")
% http://www.youtube.com/watch?v=-dhHrg-KbJ0 e to the pi i for dummies (Mathologer)

\par \url{https://en.wikipedia.org/wiki/Complex_number} (bom)
\par \url{https://pt.wikipedia.org/wiki/N\%C3\%BAmero_complexo} (ruim, cheio de erros)

\ssk

\par \Ca{2yT12} (Gabarito da P1 de 2019.2) A questão 3 usa o truque do $E$

% (c2m222srp 4 "somas-de-retangulos")
% (c2m222sra   "somas-de-retangulos")
%\par \Ca{2fT63} ``Áreas negativas não existem''

\bsk

% (find-books "__analysis/__analysis.el" "hernandez" "47" "principais identidades trigonométricas")
\par \Ca{HernandezP57} (p.47) principais identidades trigonométricas

}\anothercol{
}}

\newpage

% «resumo»  (to ".resumo")
% (c2m241ncp 3 "resumo")
% (c2m241nca   "resumo")

\def\Re{\mathsf{Re}}
\def\Im{\mathsf{Im}}
\def\Arg{\mathsf{arg}}
\def\C{\mathbb{C}}

\scalebox{0.55}{\def\colwidth{13cm}\firstcol{

$\begin{array}[t]{rcll}
     a,b,c,d &∈& \R \\
         z,w &∈& \C \\
           θ &∈& \R                         & \text{(ângulo)} \\
           k &∈& \Z                         \\ \\
   \Re(a+bi) &=& a                          & \text{(parte real)}  \\
   \Im(a+bi) &=& b                          & \text{(parte imaginária)} \\
           z &=& \Re(z) + \Im(z)i           & \text{(isto sempre vale)} \\
     \ovl{z} &=& \Re(z) - \Im(z)i           & \text{(conjugado: definição fácil)} \\
  \ovl{a+bi} &=& a - bi                     & \text{(conjugado: definição difícil)} \\
         |z| &=& \sqrt{\Re(z)^2 + \Im(z)^2} & \text{(módulo/norma: definição fácil)} \\
      |a+bi| &=& \sqrt{a^2 + b^2}           & \text{(módulo/norma: definição difícil)} \\
  \\
        180° &=& π                          & \text{($←$ lembre)} \\
          1° &=& \frac{π}{180}              & \text{($←$ lembre)} \\
         42° &=& 42\frac{π}{180}            \\
         {}° &=& \frac{π}{180}              & \text{(podemos tratar o ${}°$ como uma constante)} \\
  \\
      e^{iθ} &=& \cosθ + i\senθ             & \text{(vamos entender isto aos poucos)} \\
           E &=& c+is                       & \text{(abreviação pra igualdade acima)} \\
  \\
           z &=& |z| \, e^{i\Arg(z)}           & \text{(vamos entender isto aos poucos)} \\
         1+i &=& |1+1i| \, e^{i\Arg(1+i)}      & \text{($←$ exemplo)} \\
             &=& \sqrt{1^2+1^2} \, e^{i45°}    \\
             &=& \sqrt{2} \, e^{i\frac{π}{4}}  \\
  \end{array}
$


}\anothercol{

$\begin{array}[t]{rcll}
  (a+bi)(c+di) &=& a(c+di) + bi(c+di) \\
               &=& ac+adi + bic+bidi \\
               &=& ac+adi + bci+bd\ColorRed{(i^2)} \\
               &=& ac+adi + bci+bd\ColorRed{(-1)} \\
               &=& ac+adi + bci-bd \\
               &=& ac-bd + adi+bci \\
               &=& (ac-bd) + (ad+bc)i \\
  \\
  (ae^{iα}) (be^{iβ}) &=& (ab)(e^{iα} \, e^{iβ}) \\
                      &=& (ab)(e^{iα+iβ}) \\
                      &=& (ab)(e^{i(α+β)}) \\
                      &=& (ab)(e^{i(α+β)}) \\
  \end{array}
$

}}


\newpage

% «partes-de-cima»  (to ".partes-de-cima")
% (c2m241ncp 4 "partes-de-cima")
% (c2m241nca   "partes-de-cima")

{\bf ``Partes de cima''}

\def\ccos{\operatorname{ccos}}
\def\csen{\operatorname{csen}}
\def\eio  {e^{iθ}}
\def\eiko {e^{ikθ}}
\def\emio {e^{-iθ}}
\def\emiko{e^{-ikθ}}
\def\co   {\cos  θ}
\def\cmo  {\cos -θ}
\def\cko  {\cos kθ}
\def\so   {\sen  θ}
\def\smo  {\sen -θ}
\def\sko  {\sen kθ}
\def\Em   {E^{-1}}
\def\Emk  {E^{-k}}
\def\Ek   {E^k}


\scalebox{0.45}{\def\colwidth{15cm}\firstcol{

Fórmulas e definições:

\bsk

$\begin{array}[t]{rcl}
  \eio         &=& \co  + i\so \\
  \eiko        &=& \cko + i\sko \\
  \emio        &=& \cmo + i\smo \\
               &=& \co  + i(-\so) \\
               &=& \co  - i(\so) \\
  \eio + \emio &=& \co  + i\so \\
               &+& \co  - i\so \\
               &=& 2\co \\
  \eio - \emio &=& \co  + i\so \\
               &-& (\co  - i\so) \\
               &=& 2i\so \\
  \D\frac{\eio  + \emio}{2} &=& \co \\
  \D\frac{\eio  - \emio}{2i} &=& \so \\
  \\[-5pt]
  \D\frac{\eiko + \emiko}{2} &=& \cko \\
  \D\frac{\eiko - \emiko}{2i} &=& \sko \\
  \\[-5pt]
  \ColorRed{\ccos θ} &=& \eio + \emio \\
  \ColorRed{\csen θ} &=& \eio - \emio \\
  \ColorRed{\ccos kθ} &=& \eiko + \emiko \\
  \ColorRed{\csen kθ} &=& \eiko - \emiko \\
  \\
  \end{array}
  \qquad
  \begin{array}[t]{rcl}
  E            &=& c  + is \\
  \eiko        &=& \cko + i\sko \\
  \Em          &=& \cmo + i\smo \\
               &=& c  + i(-s) \\
               &=& c  - i(s) \\
  E + \Em      &=& c  + is \\
               &+& c  - is \\
               &=& 2c \\
  E - \Em      &=& c  + is \\
               &-& (c  - is) \\
               &=& 2is \\
  \D\frac{E + \Em}{2} &=& c \\
  \D\frac{E - \Em}{2i} &=& s \\
  \\[-5pt]
  \D\frac{\Ek + \Emk}{2} &=& \cko \\
  \D\frac{\Ek - \Emk}{2i} &=& \sko \\
  \\[-5pt]
  \ColorRed{\ccos θ} &=& E + \Em \\
  \ColorRed{\csen θ} &=& E - \Em \\
  \ColorRed{\ccos kθ} &=& \Ek + \Emk \\
  \ColorRed{\csen kθ} &=& \Ek - \Emk \\
  \\
  \end{array}
$

\ssk

O seno e o cosseno ``são'' frações.

O \standout{c}sen é a ``\ColorRed{parte de cima}'' do seno.

O \standout{c}cos é a ``\ColorRed{parte de cima}'' do cosseno.

}\anothercol{

Um exemplo do método:

\bsk

$\begin{array}[t]{rcl}
 (\cosθ)^3 &=& (\frac12 \ccosθ)^3 \\
           &=& (\frac12)^3 (\ccosθ)^3 \\
 \\[-5pt]
 (\ccosθ)^3 &=& (E+E^{-1})^3 \\
            &=& E^3 + 3E + 3\Em + E^{-3} \\
            &=& (E^3 + E^{-3}) + (3E + 3\Em) \\
            &=& \ccos 3θ + 3\ccosθ \\
 \\[-5pt]
 (\cosθ)^3 &=& (\frac12)^3 (\ccosθ)^3 \\
           &=& (\frac12)^3 (\ccos 3θ + 3\ccosθ) \\
           &=& \frac14 (\frac12\ccos 3θ + 3\frac12\ccosθ) \\
           &=& \frac14 (\cos 3θ + 3\cosθ) \\
 \end{array}
$

\bsk

Pra mim a parte do meio é a parte legal

das contas, e as partes de cima e de baixo

são as partes chatas (por causa das frações).

\msk

Compare com o gabarito da questão 3 daqui:

\par \Ca{2yT12} (Gabarito da P1 de 2019.2)



\bsk
\bsk
\bsk
\bsk

{\bf Exercício}

Use a técnica acima pra integrar:

a) $\intth{(\cosθ)^2}$

b) $\intth{(\senθ)^2}$

c) $\intth{(\senθ)(\cosθ)}$

d) $\intth{(\sen 2θ)(\cos 3θ)}$

}}



\newpage

% «dots»  (to ".dots")
% (c2m241ncp 5 "dots")
% (c2m241nca   "dots")
% (find-es "maxima" "2024.1-intro-complex")

%M (%i1) as_33 : create_list(x+%i*y, y, seqn(2,0,2), x, seqn(0,2,2));
%M (%o1) \left[ 2\,i , 2\,i+1 , 2\,i+2 , i , i+1 , i+2 , 0 , 1 , 2 \right] 
%M (%i2) as_55 : create_list(x+%i*y, y, seqn(2,0,4), x, seqn(0,2,4))$
%M 
%M (%i3) as    : as_33;
%M (%o3) \left[ 2\,i , 2\,i+1 , 2\,i+2 , i , i+1 , i+2 , 0 , 1 , 2 \right] 
%M (%i4) xyrange(r) := [xr(-r,r), yr(-r,r), more(proportional_axes=xy)];
%M (%o4) \mathrm{xyrange}\left(r\right):=\left[ \mathrm{xr}\left(-r , r\right) , \mathrm{yr}\left(-r , r\right) , \mathrm{more}\left(\mathrm{proportional\_axes}=\mathrm{xy}\right) \right] 
%M (%i5) myqdraw_nhr(n,h,r,[rest]) :=
%M   myQdraw(format("complex-~a",n), format("height=~acm",h), xyrange(r), rest);
%M (%i6)  myqdraw_nhr(1,5,4, zpts(as_33,    myps(4),pc(red)));
%M (%o6) \includegraphics[height=5cm]{2024-1-C2/complex-1.pdf}
%M (%i7) myqdraw_nhr(2,5,4, zpts(as_55,    myps(4),pc(orange)));
%M (%o7) \includegraphics[height=5cm]{2024-1-C2/complex-2.pdf}
%L maximahead:sa("dots", "")
\pu

%M (%i8) myqdraw_nhr(3,5,4, zpts(as_55+1,  myps(4),pc(orange)));
%M (%o8) \includegraphics[height=5cm]{2024-1-C2/complex-3.pdf}
%M (%i9) myqdraw_nhr(4,5,4, zpts(as_55+%i, myps(4),pc(orange)));
%M (%o9) \includegraphics[height=5cm]{2024-1-C2/complex-4.pdf}
%M (%i10) myqdraw_nhr(5,5,4, zpts(as_55*2,  myps(4),pc(orange)));
%M (%o10) \includegraphics[height=5cm]{2024-1-C2/complex-5.pdf}
%M (%i11) myqdraw_nhr(6,5,4, zpts(as_55*(1+%i), myps(4),pc(orange)));
%M (%o11) \includegraphics[height=5cm]{2024-1-C2/complex-6.pdf}
%M (%i12) 
%L maximahead:sa("dots 2", "")
\pu

\scalebox{0.3}{\def\colwidth{18cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots 2}

}}


\newpage

% «dots-2»  (to ".dots-2")
% (c2m241ncp 6 "dots-2")
% (c2m241nca   "dots-2")

%M (%i12) asq_33 : makelist(z^2, z, as_33)$
%M (%i13) asq_55 : makelist(z^2, z, as_55)$
%M (%i14) myqdraw_nhr(7,5,8, zpts(asq_33, myps(4),pc(red)));
%M (%o14) \includegraphics[height=5cm]{2024-1-C2/complex-7.pdf}
%M (%i15) myqdraw_nhr(8,5,8, zpts(asq_55, myps(1),pc(red)));
%M (%o15) \includegraphics[height=5cm]{2024-1-C2/complex-8.pdf}
%L maximahead:sa("dots sq", "")
\pu

%M (%i16) as_22   : create_list(x+%i*y, y, seqn(0,1,1), x, seqn(0,1,1))$
%M (%i17) as_332  : create_list(z+w,    z, as_33, w, as_22*0.2)$
%M (%i18) asq_332 : makelist(z^2, z, as_332)$
%M (%i19) myqdraw_nhr( 9,5,10,  zpts(as_332, myps(1),pc(red)));
%M (%o19) \includegraphics[height=5cm]{2024-1-C2/complex-9.pdf}
%M (%i20) myqdraw_nhr(10,5,10, zpts(asq_332, myps(1),pc(red)));
%M (%o20) \includegraphics[height=5cm]{2024-1-C2/complex-10.pdf}
%M (%i21) 
%L maximahead:sa("dots sq 2", "")
\pu

\scalebox{0.4}{\def\colwidth{13cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots sq}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots sq 2}

}}



\newpage

% «maxima»  (to ".maxima")
% 2hT233 (c2m232ncp 5 "maxima")
%        (c2m232nca   "maxima")
%        (find-es "maxima" "2023-2-C2-laurent2")

%M (%i1) p : 4*x^2 + 5*x^1 + 6*x^0 + 7*x^-1 + 8*x^-2;
%M (%o1) 4\,x^2+5\,x+{\frac{7}{x}}+{\frac{8}{x^2}}+6
%M (%i2) q : 4*E^2 + 5*E^1 + 6*E^0 + 7*E^-1;
%M (%o2) 4\,E^2+5\,E+{\frac{7}{E}}+6
%M (%i3) lpdot(p, x);
%M (%o3) \begin{pmatrix}4&5&6&\mbox{ . }&7&8\cr \end{pmatrix}
%M (%i4) lpdot(q, E);
%M (%o4) \begin{pmatrix}4&5&6&\mbox{ . }&7\cr \end{pmatrix}
%M (%i5) f :  cos(th)^3;
%M (%o5) \left(\cos \theta \right)^3
%M (%i6) g : ccos(th)^3;
%M (%o6) 8\,\left(\cos \theta \right)^3
%M (%i7) lpe(f);
%M (%o7) {\frac{\cos \left(3\,\theta \right)}{4}}+{\frac{3\,\cos \theta }{4}}
%M (%i8) lpe(g);
%M (%o8) 2\,\cos \left(3\,\theta \right)+6\,\cos \theta 
%M (%i9) 
%L maximahead:sa("laurent2 a", "")
\pu

% «maxima-2»  (to ".maxima-2")
% (c2m232ncp 6 "maxima-2")
% (c2m232nca   "maxima-2")

%M (%i9)                         exponentialize(f);
%M (%o9) {\frac{\left(e^{i\,\theta }+e^ {- i\,\theta  }\right)^3}{8}}
%M (%i10)                 expand(exponentialize(f));
%M (%o10) {\frac{e^{3\,i\,\theta }}{8}}+{\frac{3\,e^{i\,\theta }}{8}}+{\frac{3\,e^ {- i\,\theta  }}{8}}+{\frac{e^ {- 3\,i\,\theta  }}{8}}
%M (%i11)        demoivre(expand(exponentialize(f)));
%M (%o11) {\frac{i\,\sin \left(3\,\theta \right)+\cos \left(3\,\theta \right)}{8}}+{\frac{\cos \left(3\,\theta \right)-i\,\sin \left(3\,\theta \right)}{8}}+{\frac{3\,\left(i\,\sin \theta +\cos \theta \right)}{8}}+{\frac{3\,\left(\cos \theta -i\,\sin \theta \right)}{8}}
%M (%i12) expand(demoivre(expand(exponentialize(f))));
%M (%o12) {\frac{\cos \left(3\,\theta \right)}{4}}+{\frac{3\,\cos \theta }{4}}
%M (%i13)      subst(th_E,expand(exponentialize(f)));
%M (%o13) {\frac{E^3}{8}}+{\frac{3\,E}{8}}+{\frac{3}{8\,E}}+{\frac{1}{8\,E^3}}
%M (%i14)      subst(th_E,expand(exponentialize(g)));
%M (%o14) E^3+3\,E+{\frac{3}{E}}+{\frac{1}{E^3}}
%M (%i15) 
%M lpE(f);
%M (%o15) \begin{pmatrix}{\frac{1}{8}}&0&{\frac{3}{8}}&0&\mbox{ . }&{\frac{3}{8}}&0&{\frac{1}{8}}\cr \end{pmatrix}
%M (%i16) lpE(g);
%M (%o16) \begin{pmatrix}1&0&3&0&\mbox{ . }&3&0&1\cr \end{pmatrix}
%L maximahead:sa("laurent2 b", "")

%M (%i17) lpE(  ccos(th)^3);
%M (%o17) \begin{pmatrix}1&0&3&0&\mbox{ . }&3&0&1\cr \end{pmatrix}
%M (%i18) lpE(  ccos(th));
%M (%o18) \begin{pmatrix}1&0&\mbox{ . }&1\cr \end{pmatrix}
%M (%i19) lpE(3*ccos(th));
%M (%o19) \begin{pmatrix}3&0&\mbox{ . }&3\cr \end{pmatrix}
%M (%i20) lpE(ccos(3*th));
%M (%o20) \begin{pmatrix}1&0&0&0&\mbox{ . }&0&0&1\cr \end{pmatrix}
%M (%i21) lpE(ccos(3*th)+3*ccos(th));
%M (%o21) \begin{pmatrix}1&0&3&0&\mbox{ . }&3&0&1\cr \end{pmatrix}
%M (%i22) 
%M lpE( ccos(th)   );
%M (%o22) \begin{pmatrix}1&0&\mbox{ . }&1\cr \end{pmatrix}
%M (%i23) lpE( ccos(th)^2 );
%M (%o23) \begin{pmatrix}1&0&2&\mbox{ . }&0&1\cr \end{pmatrix}
%L maximahead:sa("laurent2 c", "")
\pu

%M (%i24) lpE( ccos(th)^3 );
%M (%o24) \begin{pmatrix}1&0&3&0&\mbox{ . }&3&0&1\cr \end{pmatrix}
%M (%i25) lpE( csin(th)     );
%M (%o25) \begin{pmatrix}1&0&\mbox{ . }&-1\cr \end{pmatrix}
%M (%i26) lpE( csin(th)^2   );
%M (%o26) \begin{pmatrix}1&0&-2&\mbox{ . }&0&1\cr \end{pmatrix}
%M (%i27) lpE( csin(th)^3   );
%M (%o27) \begin{pmatrix}1&0&-3&0&\mbox{ . }&3&0&-1\cr \end{pmatrix}
%M (%i28) lpE( csin(2*th)   );
%M (%o28) \begin{pmatrix}1&0&0&\mbox{ . }&0&-1\cr \end{pmatrix}
%M (%i29) lpE( csin(2*th)^2 );
%M (%o29) \begin{pmatrix}1&0&0&0&-2&\mbox{ . }&0&0&0&1\cr \end{pmatrix}
%M (%i30) 
%L maximahead:sa("laurent2 d", "")
\pu

\scalebox{0.4}{\def\colwidth{12cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{laurent2 a}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {18cm}
\ga{laurent2 b}

}}

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\scalebox{0.5}{\def\colwidth{10cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {12cm}
\ga{laurent2 c}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {12cm}
\ga{laurent2 d}

}}



\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}



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