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Warning: this is an htmlized version!
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% «defs-T-and-B» (to ".defs-T-and-B")
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%L dofile "Maxima2.lua" -- (find-angg "LUA/Maxima2.lua")
\pu
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%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m241ncp 1 "title")
% (c2m241nca "title")
\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}
}}
\newpage
\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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