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% (find-LATEX "2025-1-C2-taylor.tex") % (defun c () (interactive) (find-LATEXsh "lualatex -record 2025-1-C2-taylor.tex" :end)) % (defun C () (interactive) (find-LATEXsh "lualatex 2025-1-C2-taylor.tex" "Success!!!")) % (defun D () (interactive) (find-pdf-page "~/LATEX/2025-1-C2-taylor.pdf")) % (defun d () (interactive) (find-pdftools-page "~/LATEX/2025-1-C2-taylor.pdf")) % (defun e () (interactive) (find-LATEX "2025-1-C2-taylor.tex")) % (defun o () (interactive) (find-LATEX "2023-2-C2-taylor.tex")) % (defun u () (interactive) (find-latex-upload-links "2025-1-C2-taylor")) % (defun v () (interactive) (find-2a '(e) '(d))) % (defun d0 () (interactive) (find-ebuffer "2025-1-C2-taylor.pdf")) % (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g)) % (defun oe () (interactive) (find-2a '(o) '(e))) % (code-eec-LATEX "2025-1-C2-taylor") % (find-pdf-page "~/LATEX/2025-1-C2-taylor.pdf") % (find-sh0 "cp -v ~/LATEX/2025-1-C2-taylor.pdf /tmp/") % (find-sh0 "cp -v ~/LATEX/2025-1-C2-taylor.pdf /tmp/pen/") % (find-xournalpp "/tmp/2025-1-C2-taylor.pdf") % file:///home/edrx/LATEX/2025-1-C2-taylor.pdf % file:///tmp/2025-1-C2-taylor.pdf % file:///tmp/pen/2025-1-C2-taylor.pdf % http://anggtwu.net/LATEX/2025-1-C2-taylor.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 "2025-1-C2-taylor" "2" "c2m251ta" "c2ta") % «.defs» (to "defs") % «.defs-T-and-B» (to "defs-T-and-B") % «.defs-caepro» (to "defs-caepro") % «.defs-pict2e» (to "defs-pict2e") % «.defs-maxima» (to "defs-maxima") % «.defs-V» (to "defs-V") % «.title» (to "title") % «.links» (to "links") % «.links-stewart» (to "links-stewart") % «.maxima-f_x» (to "maxima-f_x") % «.maxima-polys» (to "maxima-polys") % «.maxima-cis» (to "maxima-cis") \documentclass[oneside,12pt]{article} \usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref") \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{pict2e} \usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor") \usepackage{colorweb} % (find-es "tex" "colorweb") %\usepackage{tikz} % % (find-LATEX "dednat7-test1.tex") %\usepackage{proof} % For derivation trees ("%:" lines) %\input diagxy % For 2D diagrams ("%D" lines) %\xyoption{curve} % For the ".curve=" feature in 2D diagrams % \usepackage{edrx21} % (find-LATEX "edrx21.sty") \input edrxaccents.tex % (find-LATEX "edrxaccents.tex") \input edrx21chars.tex % (find-LATEX "edrx21chars.tex") \input edrxheadfoot.tex % (find-LATEX "edrxheadfoot.tex") \input edrxgac2.tex % (find-LATEX "edrxgac2.tex") % % (find-es "tex" "geometry") \usepackage[a6paper, landscape, top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot ]{geometry} % \begin{document} % «defs» (to ".defs") % (find-LATEX "edrx21defs.tex" "colors") % (find-LATEX "edrx21.sty") \def\drafturl{http://anggtwu.net/LATEX/2025-1-C2.pdf} \def\drafturl{http://anggtwu.net/2025.1-C2.html} \def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}} % (find-LATEX "2024-1-C2-carro.tex" "defs-caepro") % (find-LATEX "2024-1-C2-carro.tex" "defs-pict2e") \catcode`\^^J=10 \directlua{dofile "dednat7load.lua"} % (find-LATEX "dednat7load.lua") \directlua{dednat7preamble()} % (find-angg "LUA/DednatPreamble1.lua") \directlua{dednat7oldheads()} % (find-angg "LUA/Dednat7oldheads.lua") % «defs-T-and-B» (to ".defs-T-and-B") \long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}} \def\T(Total: #1 pts){{\bf(Total: #1)}} \def\T(Total: #1 pts){{\bf(Total: #1 pts)}} \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") \def\Caurl #1{\expr{Caurl("#1")}} \def\Cahref#1#2{\href{\Caurl{#1}}{#2}} \def\Ca #1{\Cahref{#1}{#1}} % «defs-pict2e» (to ".defs-pict2e") %L dofile "Piecewise2.lua" -- (find-LATEX "Piecewise2.lua") %L --dofile "Escadas1.lua" -- (find-LATEX "Escadas1.lua") \def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}} \def\pictaxesstyle{\linethickness{0.5pt}} \def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}} \celllower=2.5pt % «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") % (c2m251tap 1 "title") % (c2m251taa "title") \thispagestyle{empty} \begin{center} \vspace*{1.2cm} {\bf \Large Cálculo 2 - 2025.1} \bsk Aula 30: séries de Taylor \bsk Eduardo Ochs - RCN/PURO/UFF \url{http://anggtwu.net/2025.1-C2.html} \end{center} \newpage % «links» (to ".links") % (c2m251tap 2 "links") % (c2m251taa "links") {\bf Links} \scalebox{0.44}{\def\colwidth{16cm}\firstcol{ % «links-stewart» (to ".links-stewart") % (c2m251tap 2 "links-stewart") % (c2m251taa "links-stewart") \par Stewart: % (find-books "__analysis/__analysis.el" "stewart-pt" "470" "7.8 Integrais Impróprias") % (find-books "__analysis/__analysis.el" "stewart-pt" "623" "11 Sequências e Séries Infinitas") % (find-books "__analysis/__analysis.el" "stewart-pt" "624" "11.1 Sequências") % (find-books "__analysis/__analysis.el" "stewart-pt" "636" "11.2 Séries") % (find-books "__analysis/__analysis.el" "stewart-pt" "645" "11.3 O Teste da Integral e Estimativas de Somas") % (find-books "__analysis/__analysis.el" "stewart-pt" "652" "11.4 Os Testes de Comparação") % (find-books "__analysis/__analysis.el" "stewart-pt" "657" "11.5 Séries Alternadas") % (find-books "__analysis/__analysis.el" "stewart-pt" "661" "11.6 Convergência Absoluta e os Testes da Razão e da Raiz") % (find-books "__analysis/__analysis.el" "stewart-pt" "667" "11.7 Estratégia para Testes de Séries") % (find-books "__analysis/__analysis.el" "stewart-pt" "669" "11.8 Séries de Potência") % (find-books "__analysis/__analysis.el" "stewart-pt" "671" "raio de convergência") % (find-books "__analysis/__analysis.el" "stewart-pt" "674" "11.9 Representações de Funções como Séries de Potências") % (find-books "__analysis/__analysis.el" "stewart-pt" "679" "11.10 Séries de Taylor e Maclaurin") % (find-books "__analysis/__analysis.el" "stewart-pt" "681" "Figura 1") % (find-books "__analysis/__analysis.el" "stewart-pt" "681" "polinômio de Taylor de n-ésimo grau") % (find-books "__analysis/__analysis.el" "stewart-pt" "685" "em colunas") % (find-books "__analysis/__analysis.el" "stewart-pt" "687" "Tabela 1: séries de MacLaurin importantes") % (find-books "__analysis/__analysis.el" "stewart-pt" "689" "11.10 Exercícios") % (find-books "__analysis/__analysis.el" "stewart-pt" "692" "11.11 Aplicações dos Polinômios de Taylor") % (find-books "__analysis/__analysis.el" "stewart-pt" "701" "Revisão") % (find-books "__analysis/__analysis.el" "stewart-pt" "703" "Problemas Quentes") \par \Ca{StewPtCap7p55} (p.470) 7.8 Integrais Impróprias \par \Ca{StewPtCap11p5} (p.623) 11 Sequências e Séries Infinitas \par \Ca{StewPtCap11p6} (p.624) 11.1 Sequências \par \Ca{StewPtCap11p18} (p.636) 11.2 Séries \par \Ca{StewPtCap11p27} (p.645) 11.3 O Teste da Integral e Estimativas de Somas \par \Ca{StewPtCap11p34} (p.652) 11.4 Os Testes de Comparação \par \Ca{StewPtCap11p39} (p.657) 11.5 Séries Alternadas \par \Ca{StewPtCap11p43} (p.661) 11.6 Convergência Absoluta e os Testes da Razão e da Raiz \par \Ca{StewPtCap11p49} (p.667) 11.7 Estratégia para Testes de Séries \par \Ca{StewPtCap11p51} (p.669) 11.8 Séries de Potência \par \Ca{StewPtCap11p53} (p.671) ...raio de convergência \par \Ca{StewPtCap11p56} (p.674) 11.9 Representações de Funções como Séries de Potências \par \Ca{StewPtCap11p61} (p.679) 11.10 Séries de Taylor e Maclaurin \par \Ca{StewPtCap11p63} (p.681) Figura 1: quando $n$ aumenta, $T_n$ parece aproximar... \par \Ca{StewPtCap11p63} (p.681) ...{\bf polinômio de Taylor de $n$-ésimo grau} \par \Ca{StewPtCap11p67} (p.685) ...{\bf em colunas} \par \Ca{StewPtCap11p69} (p.687) Tabela 1: séries de MacLaurin importantes \par \Ca{StewPtCap11p71} (p.689) 11.10 Exercícios \par \Ca{StewPtCap11p74} (p.692) 11.11 Aplicações dos Polinômios de Taylor \par \Ca{StewPtCap11p74} (p.692) {\bf Aproximando funções por polinômios} \par \Ca{StewPtCap11p83} (p.701) Revisão \par \Ca{StewPtCap11p85} (p.703) Problemas Quentes }\anothercol{ % http://www.youtube.com/watch?v=p-LbzWmm2zk The OTHER Painter's Paradox - Gabriel's horn % (find-youtubedl-links "/sda5/videos/Math/" "Torricelli_s_2nd_paradox_and_its_14th_century_genius_monk_resolution" "p-LbzWmm2zk" ".webm" "gabrielshorn") % (find-mpv-links) % (mf) % (ms) % (code-video "gabrielshornvideo" "/sda5/videos/Math/Torricelli_s_2nd_paradox_and_its_14th_century_genius_monk_resolution-p-LbzWmm2zk.webm") % (find-gabrielshornvideo) % (find-gabrielshornvideo "0:00") \par Vídeo do Mathologer -- assista do 4:00 ao 12:21: \par \url{http://www.youtube.com/watch?v=p-LbzWmm2zk} \msk Quadros: % 2hQ32: (find-angg ".emacs" "c2q232" "11,sep19: Frações parciais") % 2hQ70: (find-angg ".emacs" "c2q232" "31,nov06: Revisão de variáveis complexas") % 2hQ75: (find-angg ".emacs" "c2q232" "34,nov13: EDOs exatas") % 2hQ89: (find-angg ".emacs" "c2q232" "39,nov22: Séries de Taylor") \par \Ca{2hQ32} (2023.2) Frações parciais \par \Ca{2hQ70} (2023.2) Revisão de variáveis complexas \par \Ca{2hQ75} (2023.2) EDOs exatas \par \Ca{2hQ89} (2023.2) Séries de Taylor \ssk % 2hT234: (c2m232ncp 6 "maxima-2") % (c2m232nca "maxima-2") \par \Ca{2hT234} Código em Maxima % Caixinhas: }} \newpage % «maxima-f_x» (to ".maxima-f_x") % (c2m251tap 3 "maxima-f_x") % (c2m251taa "maxima-f_x") %M (%i1) load("~/MAXIMA/2025-1-taylor.mac")$ %M (%i2) gradef(f (x), f_x (x))$ %M (%i3) gradef(f_x (x), f_xx (x))$ %M (%i4) gradef(f_xx (x), f_xxx (x))$ %M (%i5) gradef(f_xxx (x), f_xxxx (x))$ %M (%i6) gradef(f_xxxx(x), f_xxxxx(x))$ %M (%i7) myt : Taylor(f(x),x,0,4)$ %M (%i8) myt@@abcd(x); %M (%o8) \left[ \begin{pmatrix}f\left(x\right)\cr \mathrm{f\_x}\left(x\right)\cr \mathrm{f\_xx}\left(x\right)\cr \mathrm{f\_xxx}\left(x\right)\cr \mathrm{f\_xxxx}\left(x\right)\cr \end{pmatrix} , \begin{pmatrix}f\left(0\right)\cr \mathrm{f\_x}\left(0\right)\cr \mathrm{f\_xx}\left(0\right)\cr \mathrm{f\_xxx}\left(0\right)\cr \mathrm{f\_xxxx}\left(0\right)\cr \end{pmatrix} , \begin{pmatrix}f\left(0\right)\cr \mathrm{f\_x}\left(0\right)\cr {\frac{\mathrm{f\_xx}\left(0\right)}{2}}\cr {\frac{\mathrm{f\_xxx}\left(0\right)}{6}}\cr {\frac{\mathrm{f\_xxxx}\left(0\right)}{24}}\cr \end{pmatrix} , \begin{pmatrix}f\left(0\right)\cr \mathrm{f\_x}\left(0\right)\,x\cr {\frac{\mathrm{f\_xx}\left(0\right)\,x^2}{2}}\cr {\frac{\mathrm{f\_xxx}\left(0\right)\,x^3}{6}}\cr {\frac{\mathrm{f\_xxxx}\left(0\right)\,x^4}{24}}\cr \end{pmatrix} \right] %M (%i9) myt@@s(x); %M (%o9) {\frac{\mathrm{f\_xxxx}\left(0\right)\,x^4}{24}}+{\frac{\mathrm{f\_xxx}\left(0\right)\,x^3}{6}}+{\frac{\mathrm{f\_xx}\left(0\right)\,x^2}{2}}+\mathrm{f\_x}\left(0\right)\,x+f\left(0\right) %M (%i10) myt@@s(a); %M (%o10) {\frac{\mathrm{f\_xxxx}\left(0\right)\,a^4}{24}}+{\frac{\mathrm{f\_xxx}\left(0\right)\,a^3}{6}}+{\frac{\mathrm{f\_xx}\left(0\right)\,a^2}{2}}+\mathrm{f\_x}\left(0\right)\,a+f\left(0\right) %M (%i11) myt@@s(1/10); %M (%o11) {\frac{\mathrm{f\_xxxx}\left(0\right)}{240000}}+{\frac{\mathrm{f\_xxx}\left(0\right)}{6000}}+{\frac{\mathrm{f\_xx}\left(0\right)}{200}}+{\frac{\mathrm{f\_x}\left(0\right)}{10}}+f\left(0\right) %M (%i12) %L maximahead:sa("f_x", "") \pu \scalebox{0.5}{\def\colwidth{9cm}\firstcol{ \vspace*{0cm} \def\hboxthreewidth {14cm} \ga{f_x} }\anothercol{ }} \newpage % «maxima-polys» (to ".maxima-polys") % (c2m251tap 4 "maxima-polys") % (c2m251taa "maxima-polys") %M (%i1) load("~/MAXIMA/2025-1-taylor.mac")$ %M (%i2) f3(x) := a + b*x + c*x^2 + d*x^3$ %M (%i3) myt : Taylor(f3(x),x,0,4)$ %M (%i4) myt@@abcd(x); %M (%o4) \left[ \begin{pmatrix}d\,x^3+c\,x^2+b\,x+a\cr 3\,d\,x^2+2\,c\,x+b\cr 6\,d\,x+2\,c\cr 6\,d\cr 0\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr 2\,c\cr 6\,d\cr 0\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr c\cr d\cr 0\cr \end{pmatrix} , \begin{pmatrix}a\cr b\,x\cr c\,x^2\cr d\,x^3\cr 0\cr \end{pmatrix} \right] %M (%i5) myt@@s(x); %M (%o5) d\,x^3+c\,x^2+b\,x+a %M (%i6) f4(x) := a + b*x + c*x^2 + d*x^3 + e*x^4$ %M (%i7) myt : Taylor(f4(x),x,0,4)$ %M (%i8) myt@@abcd(x); %M (%o8) \left[ \begin{pmatrix}e\,x^4+d\,x^3+c\,x^2+b\,x+a\cr 4\,e\,x^3+3\,d\,x^2+2\,c\,x+b\cr 12\,e\,x^2+6\,d\,x+2\,c\cr 24\,e\,x+6\,d\cr 24\,e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr 2\,c\cr 6\,d\cr 24\,e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr c\cr d\cr e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\,x\cr c\,x^2\cr d\,x^3\cr e\,x^4\cr \end{pmatrix} \right] %M (%i9) myt@@s(x); %M (%o9) e\,x^4+d\,x^3+c\,x^2+b\,x+a %M (%i10) f5(x) := a + b*x + c*x^2 + d*x^3 + e*x^4 + f*x^5$ %M (%i11) myt : Taylor(f5(x),x,0,4)$ %M (%i12) myt@@abcd(x); %M (%o12) \left[ \begin{pmatrix}f\,x^5+e\,x^4+d\,x^3+c\,x^2+b\,x+a\cr 5\,f\,x^4+4\,e\,x^3+3\,d\,x^2+2\,c\,x+b\cr 20\,f\,x^3+12\,e\,x^2+6\,d\,x+2\,c\cr 60\,f\,x^2+24\,e\,x+6\,d\cr 120\,f\,x+24\,e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr 2\,c\cr 6\,d\cr 24\,e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\cr c\cr d\cr e\cr \end{pmatrix} , \begin{pmatrix}a\cr b\,x\cr c\,x^2\cr d\,x^3\cr e\,x^4\cr \end{pmatrix} \right] %M (%i13) myt@@s(x); %M (%o13) e\,x^4+d\,x^3+c\,x^2+b\,x+a %L maximahead:sa("polys", "") \pu \scalebox{0.37}{\def\colwidth{9cm}\firstcol{ \vspace*{0cm} \def\hboxthreewidth {14cm} \ga{polys} }\anothercol{ }} \newpage % «maxima-cis» (to ".maxima-cis") % (c2m251tap 5 "maxima-cis") % (c2m251taa "maxima-cis") %M (%i1) load("~/MAXIMA/2025-1-taylor.mac")$ %M (%i2) myte : Taylor(exp( th) , th,0,5)$ %M (%i3) myte2 : Taylor(exp( 2*th) , th,0,5)$ %M (%i4) mytei : Taylor(exp(%i*th) , th,0,5)$ %M (%i5) mytc : Taylor(cos(th) , th,0,5)$ %M (%i6) myts : Taylor( sin(th), th,0,5)$ %M (%i7) mytis : Taylor( %i*sin(th), th,0,5)$ %M (%i8) mytcis : Taylor(cos(th)+%i*sin(th), th,0,5)$ %M (%i9) myte @@abcd(th); %M (%o9) \left[ \begin{pmatrix}e^{\theta }\cr e^{\theta }\cr e^{\theta }\cr e^{\theta }\cr e^{\theta }\cr e^{\theta }\cr \end{pmatrix} , \begin{pmatrix}1\cr 1\cr 1\cr 1\cr 1\cr 1\cr \end{pmatrix} , \begin{pmatrix}1\cr 1\cr {\frac{1}{2}}\cr {\frac{1}{6}}\cr {\frac{1}{24}}\cr {\frac{1}{120}}\cr \end{pmatrix} , \begin{pmatrix}1\cr \theta \cr {\frac{\theta ^2}{2}}\cr {\frac{\theta ^3}{6}}\cr {\frac{\theta ^4}{24}}\cr {\frac{\theta ^5}{120}}\cr \end{pmatrix} \right] %M (%i10) myte2 @@abcd(th); %M (%o10) \left[ \begin{pmatrix}e^{2\,\theta }\cr 2\,e^{2\,\theta }\cr 4\,e^{2\,\theta }\cr 8\,e^{2\,\theta }\cr 16\,e^{2\,\theta }\cr 32\,e^{2\,\theta }\cr \end{pmatrix} , \begin{pmatrix}1\cr 2\cr 4\cr 8\cr 16\cr 32\cr \end{pmatrix} , \begin{pmatrix}1\cr 2\cr 2\cr {\frac{4}{3}}\cr {\frac{2}{3}}\cr {\frac{4}{15}}\cr \end{pmatrix} , \begin{pmatrix}1\cr 2\,\theta \cr 2\,\theta ^2\cr {\frac{4\,\theta ^3}{3}}\cr {\frac{2\,\theta ^4}{3}}\cr {\frac{4\,\theta ^5}{15}}\cr \end{pmatrix} \right] %M (%i11) mytei @@abcd(th); %M (%o11) \left[ \begin{pmatrix}e^{i\,\theta }\cr i\,e^{i\,\theta }\cr -e^{i\,\theta }\cr -\left(i\,e^{i\,\theta }\right)\cr e^{i\,\theta }\cr i\,e^{i\,\theta }\cr \end{pmatrix} , \begin{pmatrix}1\cr i\cr -1\cr -i\cr 1\cr i\cr \end{pmatrix} , \begin{pmatrix}1\cr i\cr -\left({\frac{1}{2}}\right)\cr -\left({\frac{i}{6}}\right)\cr {\frac{1}{24}}\cr {\frac{i}{120}}\cr \end{pmatrix} , \begin{pmatrix}1\cr i\,\theta \cr -\left({\frac{\theta ^2}{2}}\right)\cr -\left({\frac{i\,\theta ^3}{6}}\right)\cr {\frac{\theta ^4}{24}}\cr {\frac{i\,\theta ^5}{120}}\cr \end{pmatrix} \right] %L maximahead:sa("cis", "") \pu %M (%i12) mytc @@abcd(th); %M (%o12) \left[ \begin{pmatrix}\cos \theta \cr -\sin \theta \cr -\cos \theta \cr \sin \theta \cr \cos \theta \cr -\sin \theta \cr \end{pmatrix} , \begin{pmatrix}1\cr 0\cr -1\cr 0\cr 1\cr 0\cr \end{pmatrix} , \begin{pmatrix}1\cr 0\cr -\left({\frac{1}{2}}\right)\cr 0\cr {\frac{1}{24}}\cr 0\cr \end{pmatrix} , \begin{pmatrix}1\cr 0\cr -\left({\frac{\theta ^2}{2}}\right)\cr 0\cr {\frac{\theta ^4}{24}}\cr 0\cr \end{pmatrix} \right] %M (%i13) myts @@abcd(th); %M (%o13) \left[ \begin{pmatrix}\sin \theta \cr \cos \theta \cr -\sin \theta \cr -\cos \theta \cr \sin \theta \cr \cos \theta \cr \end{pmatrix} , \begin{pmatrix}0\cr 1\cr 0\cr -1\cr 0\cr 1\cr \end{pmatrix} , \begin{pmatrix}0\cr 1\cr 0\cr -\left({\frac{1}{6}}\right)\cr 0\cr {\frac{1}{120}}\cr \end{pmatrix} , \begin{pmatrix}0\cr \theta \cr 0\cr -\left({\frac{\theta ^3}{6}}\right)\cr 0\cr {\frac{\theta ^5}{120}}\cr \end{pmatrix} \right] %M (%i14) mytis @@abcd(th); %M (%o14) \left[ \begin{pmatrix}i\,\sin \theta \cr i\,\cos \theta \cr -\left(i\,\sin \theta \right)\cr -\left(i\,\cos \theta \right)\cr i\,\sin \theta \cr i\,\cos \theta \cr \end{pmatrix} , \begin{pmatrix}0\cr i\cr 0\cr -i\cr 0\cr i\cr \end{pmatrix} , \begin{pmatrix}0\cr i\cr 0\cr -\left({\frac{i}{6}}\right)\cr 0\cr {\frac{i}{120}}\cr \end{pmatrix} , \begin{pmatrix}0\cr i\,\theta \cr 0\cr -\left({\frac{i\,\theta ^3}{6}}\right)\cr 0\cr {\frac{i\,\theta ^5}{120}}\cr \end{pmatrix} \right] %M (%i15) mytcis@@abcd(th); %M (%o15) \left[ \begin{pmatrix}i\,\sin \theta +\cos \theta \cr i\,\cos \theta -\sin \theta \cr -\left(i\,\sin \theta \right)-\cos \theta \cr \sin \theta -i\,\cos \theta \cr i\,\sin \theta +\cos \theta \cr i\,\cos \theta -\sin \theta \cr \end{pmatrix} , \begin{pmatrix}1\cr i\cr -1\cr -i\cr 1\cr i\cr \end{pmatrix} , \begin{pmatrix}1\cr i\cr -\left({\frac{1}{2}}\right)\cr -\left({\frac{i}{6}}\right)\cr {\frac{1}{24}}\cr {\frac{i}{120}}\cr \end{pmatrix} , \begin{pmatrix}1\cr i\,\theta \cr -\left({\frac{\theta ^2}{2}}\right)\cr -\left({\frac{i\,\theta ^3}{6}}\right)\cr {\frac{\theta ^4}{24}}\cr {\frac{i\,\theta ^5}{120}}\cr \end{pmatrix} \right] %M (%i16) mytei @@s(th); %M (%o16) {\frac{i\,\theta ^5}{120}}+{\frac{\theta ^4}{24}}-{\frac{i\,\theta ^3}{6}}-{\frac{\theta ^2}{2}}+i\,\theta +1 %M (%i17) mytcis@@s(th); %M (%o17) {\frac{i\,\theta ^5}{120}}+{\frac{\theta ^4}{24}}-{\frac{i\,\theta ^3}{6}}-{\frac{\theta ^2}{2}}+i\,\theta +1 %L maximahead:sa("cis 2", "") \pu \scalebox{0.33}{\def\colwidth{14cm}\firstcol{ \vspace*{0cm} \def\hboxthreewidth {14cm} \ga{cis} }\anothercol{ \vspace*{0cm} \def\hboxthreewidth {14cm} \ga{cis 2} }} \GenericWarning{Success:}{Success!!!} % Used by `M-x cv' \end{document} % (find-pdfpages2-links "~/LATEX/" "2025-1-C2-taylor") % Local Variables: % coding: utf-8-unix % ee-tla: "c2ta" % ee-tla: "c2m251ta" % End: