% (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

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