%% $Id: firamath-otf-doc.tex 809 2018-09-17 06:58:43Z herbert $
\listfiles
\documentclass[english,log-declarations=false]{article}
\usepackage{amsmath,esvect}
\usepackage{FiraSans}
\usepackage{FiraMono}
\usepackage[fakebold]{firamath-otf}
\usepackage{babel}
\usepackage{booktabs}
\usepackage{xltabular}
\usepackage{listings}
\usepackage{xspace}
\usepackage{ctexhook}
\usepackage{physics}
\usepackage{xcolor,url}
\usepackage{varioref,multido}
\newcommand\Macro[1]{\texttt{\textbackslash#1}}
\usepackage{dtk-extern}

\newenvironment{demoquote}
               {\begingroup
                \setlength{\topsep}{0pt}
                \setlength{\partopsep}{0pt}
                \list{}{\rightmargin\leftmargin}%
                \item\relax}
               {\endlist\endgroup}

\def\testfeature#1#2#3{{\fontspec[RawFeature={+#2}]{#1}#3\relax}}


\makeatletter
\def\e@alloc#1#2#3#4#5#6{%
  \global\advance#3\@ne
  \e@ch@ck{#3}{#4}{#5}#1%
  \allocationnumber#3\relax
  \global#2#6\allocationnumber
  }
\def\@pr@videpackage[#1]{%
  \expandafter\xdef\csname ver@\@currname.\@currext\endcsname{#1}}
\def\@providesfile#1[#2]{%
    \expandafter\xdef\csname ver@#1\endcsname{#2}%
  \endgroup}
\def\@latex@info#1{}
\def\@font@info#1{}
\def\ClassInfo#1#2{}
\def\PackageInfo#1#2{}

\ExplSyntaxOn
\cs_new:Npn \__fonttest_close_msg:nn #1#2
  { \msg_redirect_name:nnn {#1} {#2} { none } }
\__fonttest_close_msg:nn { LaTeX / xparse } { not-single-char     }
% \__fonttest_close_msg:nn { fontspec       } { defining-font       }
% \__fonttest_close_msg:nn { fontspec       } { no-scripts          }
\__fonttest_close_msg:nn { unicode-math   } { patch-macro         }
\ctex_at_end_package:nn { geometry } { \def\Gm@showparams#1{} }

\unimathsetup
  {
    math-style = ISO,
    bold-style = ISO,
    mathrm     = sym
  }

\str_new:N  \l_fonttest_font_str
\str_set:Nn \l_fonttest_font_str { fira }  % Can be either fira/xits/lm
\cs_set:Npn \WIEGHT { Regular }
\cs_set:Npn \SSTY   { }
%%%%%%%%%%%%%%%%%%%%

\str_if_eq:VnTF \l_fonttest_font_str { fira }
  {
    \cs_new:Npn \__fonttest_set_fira_math:n #1
      { \setmathfont { FiraMath-\WIEGHT.otf } [ BoldFont = *, #1 ] }
    \cs_if_exist:NTF \SSTY
      {
        \__fonttest_set_fira_math:n { }
        \__fonttest_set_fira_math:n { version = pnum,       Numbers = Proportional }
        \__fonttest_set_fira_math:n { version = upintegral, StylisticSet = 1       }
        \__fonttest_set_fira_math:n { version = hbar,       StylisticSet = 2       }
        \__fonttest_set_fira_math:n { version = complement, StylisticSet = 3       }
      }
      {
        \__fonttest_set_fira_math:n { }
        \__fonttest_set_fira_math:n { version = pnum       }
        \__fonttest_set_fira_math:n { version = upintegral }
        \__fonttest_set_fira_math:n { version = hbar       }
        \__fonttest_set_fira_math:n { version = complement }
      }
    \newfontface\firatext{FiraMath-\WIEGHT.otf}[BoldFont = *]
  }
  {
    \str_if_eq:VnTF \l_fonttest_font_str { xits }
      {
        \setmathfont { XITS~ Math }
        \setmathfont { XITS~ Math } [ BoldFont = *,                    version = pnum       ]
        \setmathfont { XITS~ Math } [ BoldFont = *, StylisticSet = 8,  version = upintegral ]
        \setmathfont { XITS~ Math } [ BoldFont = *, StylisticSet = 10, version = hbar       ]
        \setmathfont { XITS~ Math } [ BoldFont = *,                    version = complement ]
        \newfontface \firatext { XITS~ Math } [ BoldFont = * ]
      }
      {
        \str_if_eq:VnT \l_fonttest_font_str { lm }
          {
            \setmathfont { Latin~ Modern~ Math }
            \setmathfont { Latin~ Modern~ Math } [ BoldFont = *, version = pnum       ]
            \setmathfont { Latin~ Modern~ Math } [ BoldFont = *, version = upintegral ]
            \setmathfont { Latin~ Modern~ Math } [ BoldFont = *, version = hbar       ]
            \setmathfont { Latin~ Modern~ Math } [ BoldFont = *, version = complement ]
            \newfontface \firatext { Latin~ Modern~ Math } [ BoldFont = * ]
          }
      }
  }

\cs_set:Npn \LatinAlphabets { ABCDEFGHIJKLMNOPQRSTUVWXYZ }
\cs_set:Npn \latinAlphabets { abcdefghijklmnopqrstuvwxyz }

% ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
% αβγδεζηθικλμνξοπρστυφχψω
\cs_set:Npn \GreekAlphabets
  {
    \Alpha      \Beta       \Gamma      \Delta      \Epsilon
    \Zeta       \Eta        \Theta      \varTheta   \Iota
    \Kappa      \Lambda     \Mu         \Nu         \Xi
    \Omicron    \Pi         \Rho        \Sigma      \Tau
    \Upsilon    \Phi        \Chi        \Psi        \Omega
  }
\cs_set:Npn \greekAlphabets
  {
    \alpha      \beta       \gamma      \delta      \epsilon
    \varepsilon \zeta       \eta        \theta      \vartheta 
    \iota       \kappa      \varkappa   \lambda     \mu
    \nu         \xi         \omicron    \pi         \rho
    \varrho     \sigma      \varsigma   \tau        \upsilon
    \phi        \varphi     \chi        \psi        \omega
  }

% More characters.
\AtBeginDocument{
  \__um_sym:nnn { "0323 } { \underdot             } { \mathbotaccent }
  \__um_sym:nnn { "0324 } { \twounderdot          } { \mathbotaccent }
  \__um_sym:nnn { "20D3 } { \shortvertoverlay     } { \mathaccent    }
  \__um_sym:nnn { "20D6 } { \cev                  } { \mathaccent    }
  \__um_sym:nnn { "20E1 } { \leftrightarrowaccent } { \mathaccent    }
  \__um_sym:nnn { "20EC } { \underrightharpoon    } { \mathbotaccent }
  \__um_sym:nnn { "20ED } { \underleftharpoon     } { \mathbotaccent }
  \__um_sym:nnn { "20EE } { \underleftarrow       } { \mathbotaccent }
  \__um_sym:nnn { "20EF } { \underrightarrow      } { \mathbotaccent }
}

\tl_const:Nn \c__fonttest_accents_tl
  {
    % accent
    \acute \grave \check \hat \bar \breve
    % dot
    \mathring \dot \ddot \dddot \ddddot
    % arrow
    \cev \vec \leftrightarrowaccent \leftharpoonaccent \rightharpoonaccent
    % other
    \tilde \asteraccent \vertoverlay \shortvertoverlay \annuity
    % long accent
    \widehat \widetilde \widebridgeabove
    % under
    \underdot \twounderdot \threeunderdot \underleftharpoon \underrightharpoon
    \underleftarrow \underrightarrow
  }

\cs_set:Npn \MatrixII
  {
    a & b & c & d \\
    x & y & z & w
  }
\cs_set:Npn \MatrixIII
  {
    a & b & c & d \\
    k & l & m & n \\
    x & y & z & w
  }
\cs_set:Npn \MatrixIV
  {
    a & b & c & d \\
    k & l & m & n \\
    p & q & s & t \\
    x & y & z & w
  }

\NewDocumentCommand \TopAccentMap { m m }
  { \fonttest_top_accent_map:Nx #1 {#2} }
\cs_new:Npn \fonttest_top_accent_map:Nn #1#2
  { \tl_map_inline:nn {#2} { \[ \__fonttest_top_accent:n { #1 {##1} } \] } }
\cs_generate_variant:Nn \fonttest_top_accent_map:Nn { Nx }
\cs_new:Npn \__fonttest_top_accent:n #1
  { \tl_map_inline:Nn \c__fonttest_accents_tl { ##1 {#1} \, } }

\cs_set:Npn \OverUnderline #1
  {
    #1{} \quad #1{b} \quad #1{ab} \quad #1{abc} \quad #1{abcd} \quad #1{abcde} \quad #1{a+b+c}
  }
\cs_set:Npn \ListText
  { x\sb{1}, \, x\sb{2}, \, \ldots, \, x\sb{n} }
\cs_set:Npn \LigatureText
  { ff \quad fi \quad fl \quad ffi \quad ffl }

\NewDocumentCommand \PrintRadical { m m m }
  { \fonttest_print_root:nnn {#1} {#2} {#3} }
\cs_new_protected:Npn \fonttest_print_root:nnn #1#2#3
  {
    \tl_set:Nn \l__fonttest_root_tl {#2}
    \int_step_inline:nn {#3}
      {
        \tl_set:Nx \l__fonttest_root_tl
          { \exp_not:n {#1} { \exp_not:V \l__fonttest_root_tl } }
      }
    \tl_use:N \l__fonttest_root_tl
  }
\tl_new:N \l__fonttest_root_tl

\NewDocumentCommand \PrintDelimiters { m m }
  { \fonttest_print_delimiters:nnnnn {#1} {#2} { 9 } { 1.8 } { 40 } }
\cs_new_protected:Npn \fonttest_print_delimiters:nnnnn #1#2#3#4#5
  {
    \cs_set:Npn \__fonttest_left_delimiter:n ##1
      { \left #1 \vbox_to_ht:nn { ##1 pt } { } }
    \cs_set:Npn \__fonttest_right_delimiter:n ##1
      { \vbox_to_ht:nn { ##1 pt } { } \right #2 }
    \tl_set:Nx \l__fonttest_delimiter_tl
      {
        \fp_step_function:nnnN {#5} { - #4 } {#3} \__fonttest_left_delimiter:n
        #1
      }
    \tl_set:Nx \l__fonttest_delimiter_tl
      {
        \l__fonttest_delimiter_tl
        #2
        \fp_step_function:nnnN {#3} {#4} {#5} \__fonttest_right_delimiter:n
      }
    \tl_use:N \l__fonttest_delimiter_tl
  }
\tl_new:N \l__fonttest_delimiter_tl

\ExplSyntaxOff



\renewcommand\familydefault{\sfdefault}



\title{OpenType math font Fira}
\author{Herbert Voß}
\usepackage{parskip}
\parindent=0pt


\begin{document}
\maketitle

\tableofcontents


\begin{abstract}
The math font FIRA is derived from the Fira Sans and Fira Go sans serif.
There are several math versions available (\url{https://github.com/Stone-Zeng/FiraMath/}) but
only the regular version has from todays update all symbols. 
\end{abstract}


\section{Usage}

\begin{verbatim}
\usepackage[<options>]{firamath-otf}
\end{verbatim}

Optional arguments are 

\begin{description}
\item[\texttt{fakebold}] Use faked bold symbols 
\item[\texttt{usefilenames}] Use filenames for the fonts instead of the symbolic font names
\end{description}


The package itself loads by default

\begin{verbatim}
\RequirePackage{ifxetex,ifluatex,xkeyval,textcomp}
\RequirePackage{unicode-math}
\end{verbatim}




\section{The default regular weight}

\def\Q#1#2{\frac{\uppartial #1}{\uppartial #2}}
\def\half{\frac{1}{2}}
\def\vvec#1{\vv{#1}}
\newcommand\uppartial{\symup{\partial}}
\newcommand*\diff{\mathop{}\!\symup{d}}
\newcommand*\<{\negthickspace}
\newcommand*\TT{{\setBold\symup{T}\unsetBold}}
\def\DD{{\setBold\symup{D}\unsetBold}}


\subsection{Version normal}

\begin{align}
\begin{aligned}
  \Q{\varrho}{t}+\symup{div}(\varrho\vec{v}) &= 0 \\
  \varrho\Q{\vec{v}}{t}+(\varrho\vec{v}\cdot\nabla)\vec{v}   &= \vec{f}_0+\symup{div}\TT=\vec{f}_0
	-\symup{grad}p+\symup{div}\TT' \\
  \varrho T\frac{\diff s}{\diff t}               &= \varrho\frac{\diff e}{\diff t}
    -\frac{p}{\varrho}\frac{\diff\varrho}{\diff t}=-\symup{div}\vec{q}+\TT':\DD 
\end{aligned}
\end{align}

\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\TT\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\TT\right)\diff^2A}$}.\nonumber
\end{align}


\subsection{Version bold}

The bold characters are created with the optional argument \texttt{fakebold} which loads the
package \texttt{xfakebold} which writes some information into the created PDF to get bold
characters. For more informations see the documentation of \texttt{xfakebold}.

\setBold
\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\symup{T}\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\symup{T}\right)\diff^2A}$}.\nonumber
\end{align}
\unsetBold


\iffalse
\section{The thin  weight}

\setsansfont{FiraSans-Thin.otf}[BoldFont=FiraSans-SemiBold.otf]
\setmathfont{FiraMath-Thin.otf}

\subsection{Version normal}

\begin{align}
\begin{aligned}
  \Q{\varrho}{t}+\symup{div}(\varrho\vec{v}) &= 0 \\
  \varrho\Q{\vec{v}}{t}+(\varrho\vec{v}\cdot\nabla)\vec{v}   &= \vec{f}_0+\symup{div}\TT=\vec{f}_0
	-\symup{grad}p+\symup{div}\TT' \\
  \varrho T\frac{\diff s}{\diff t}               &= \varrho\frac{\diff e}{\diff t}
    -\frac{p}{\varrho}\frac{\diff\varrho}{\diff t}=-\symup{div}\vec{q}+\TT':\DD 
\end{aligned}
\end{align}

\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\TT\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\TT\right)\diff^2A}$}.\nonumber
\end{align}

\subsection{Version bold}



\setBold
\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\symup{T}\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\symup{T}\right)\diff^2A}$}.\nonumber
\end{align}
\unsetBold



\section{The light  weight}

\setsansfont{FiraSans-Light.otf}[BoldFont=FiraSans-SemiBold.otf]
\setmathfont{FiraMath-Light.otf}

\subsection{Version normal}

\begin{align}
\begin{aligned}
  \Q{\varrho}{t}+\symup{div}(\varrho\vec{v}) &= 0 \\
  \varrho\Q{\vec{v}}{t}+(\varrho\vec{v}\cdot\nabla)\vec{v}   &= \vec{f}_0+\symup{div}\TT=\vec{f}_0
	-\symup{grad}p+\symup{div}\TT' \\
  \varrho T\frac{\diff s}{\diff t}               &= \varrho\frac{\diff e}{\diff t}
    -\frac{p}{\varrho}\frac{\diff\varrho}{\diff t}=-\symup{div}\vec{q}+\TT':\DD 
\end{aligned}
\end{align}

\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\TT\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\TT\right)\diff^2A}$}.\nonumber
\end{align}

\subsection{Bold version with optional argument \texttt{fakebold}}


\setBold
\begin{align}
 \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}ec{n})\diff^2A          &= 0\\
 \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &=
        \iiint\<f_0\diff^3V+\oiint\vec{n}\cdot\symup{T}\diff^2A \\
 \Q{}{t}\iiint\<\left(\half v^2+e\right)\varrho\diff^3V+\oiint\left(\half v^2+e\right)
        \varrho\left(\vec{v}\cdot\vec{n}\,\right)\diff^2A                                & =\\
 \multispan2{\hfill${\displaystyle-\oiint\left(\vec{q}\cdot\vec{v}ec{n}\right)\diff^2A+
         \iiint\<\left(\vec{v}\cdot\vec{f}_0\right)\diff^3V+\oiint\left(\vec{v}
         \cdot\vec{n}~\symup{T}\right)\diff^2A}$}.\nonumber
\end{align}
\unsetBold


\fi


\section{Examples}

\subsection{Digits}

\begin{itemize}
  \item Digits:
        \[ 0123456789 \]
  \item Proportional digits:
        \begingroup
          \mathversion{pnum}
          \[ 0123456789 \]
        \endgroup
  \item Bold digits (\verb|\symbf|):
        \[ \symbf{0123456789} \]
  \item Bold proportional digits (\verb|\symbf|):
        \begingroup
          \mathversion{pnum}
          \[ \symbf{0123456789} \]
        \endgroup
\end{itemize}

\subsection{Alphabets}

\begin{itemize}
  \item Latin letters (mathnormal):
        \[ \LatinAlphabets \latinAlphabets \]

  \item Latin upright letters (\verb|\symup|):
        \[ \symup{\LatinAlphabets} \symup{\latinAlphabets}\]

  \item Latin typewriter letters (\verb|\symtt|):
        \[ \symtt{\LatinAlphabets} \symtt{\latinAlphabets} \]
  \item Latin bold letters (\verb|\symbf|):
        \[ \symbf{\LatinAlphabets} \symbf{\latinAlphabets}\]
  \item Latin bold upright letters (\verb|\symbfup|):
        \[ \symbfup{\LatinAlphabets}  \symbfup{\latinAlphabets} \]
  \item Latin blackboard letters (\verb|\symbb|):
        \[ \symbb{\LatinAlphabets} \symbb{\latinAlphabets}\]
        % \[ \symbb{The\ quick\ brown\ fox\ jumps\ over\ the\ lazy\ dog.} \]
        % \[ \symbb{The\ Quick\ Brown\ Fox\ Jumps\ Over\ The\ Lazy\ Dog.} \]
        % \[ \symbb{THE\ QUICK\ BROWN\ FOX\ JUMPS\ OVER\ THE\ LAZY\ DOG.} \]
  \item Greek letters:
        \[ \GreekAlphabets \greekAlphabets \]
  \item Greek upright letters (\verb|\symup|):
        \[ \symup{\GreekAlphabets} \symup{\greekAlphabets}\]
  \item Greek bold letters (\verb|\symbf|):
        \[ \symbf{\GreekAlphabets}\symbf{\greekAlphabets} \]
  \item Greek bold upright letters (\verb|\symbfup|):
        \[ \symbfup{\GreekAlphabets} \symbfup{\greekAlphabets} \]
  \item Dotless letters:
        \[ \imath + \jmath + \symup{\imath} + \symup{\jmath} \]
        % \TopAccentMap{\symbf}{\imath\jmath}
        % \TopAccentMap{\symup}{\imath\jmath}
  \item Hebrew
        \[ \aleph + \beth + \gimel + \daleth \]
  \item Ligature (text):
        {\firatext\LigatureText}
  \item Non-ligature (math):
        \[ \LigatureText + \symit{\LigatureText} + \symrm{\LigatureText} \]
  \item Miscellaneous:
        \[
          \hslash
          + \mbox{\mathversion{hbar}$\hslash$}
          + \Angstrom
        \]
        \[ \forall x > x_0, \, \exists \delta, \delta \in \varnothing \]
\end{itemize}

\subsection{Equations test}

\begin{itemize}
  \item Basic:
        \[ 1 + 2 - 3 \times 4 \div 5 \pm 6 \mp 7 \dotplus 8 = -a \oplus b \otimes c \]
  \item Binary relations
        \[ x + - \oplus \otimes \ominus \odot \oslash \cdot \cdotp \times \div y \]
  \item Set theory
        \[ A \cap B \cup C \sqcap D \sqcup R \cupleftarrow k \cupdot l \uplus m \]
        \[
            A \subset B \supset C \subseteq D \supseteq E \Subset F \Supset G
          + A \sqsubset B \sqsupset C \sqsubseteq D \sqsupseteq E
        \]
        \[
          \complement_U A \cup \complement_C C
          \subset \mbox{\mathversion{complement}$\complement_U A \cup \complement_C C$}
          \in R \smallin Q \ni Z \smallni N
        \]
  \item Superscript and subscript:
        \[ 2^2 + 2^{2^2} + 2^{2^{2^2}} + {2^2}^2 + x_a + x_{a_i} + x_{a_{i_1}} \]
  \item Arrows:
        \[
          x \leftarrow  y   \rightarrow  z \leftrightarrow  w
            \nleftarrow y   \nrightarrow z \nleftrightarrow w
            \Leftarrow  a = \Rightarrow  b \Leftrightarrow  c
            \nLeftarrow a = \nRightarrow b \nLeftrightarrow c
        \]
        \[
          x \uparrow   y \downarrow  z \updownarrow    w
            \Uparrow   a \Downarrow  b \Updownarrow    c
        \]
        \[
          p \nwarrow p \nearrow p \searrow p \swarrow p
            \Nwarrow p \Nearrow p \Searrow p \Swarrow p
        \]
        \[
          x \leftharpoonup    x \leftharpoondown  x
            \upharpoonright   x \upharpoonleft    x
            \rightharpoonup   x \rightharpoondown x
            \downharpoonright x \downharpoonleft  x
        \]
        \[
          A \longleftarrow B   \longrightarrow C \longleftrightarrow D
            \Longleftarrow E = \Longrightarrow F \Longleftrightarrow G
        \]
        \[
          X \mapsfrom Y \mapsto Z \mapsup W \mapsdown P \Mapsfrom S \Mapsto R
        \]
        \[
          M \longmapsfrom N \longmapsto O \Longmapsfrom K \Longmapsto L
        \]
        \[
          f \rightleftarrows  f \updownarrows f \leftrightarrows f \downuparrows
          g \rightrightarrows g \upuparrows   g \leftleftarrows  g \downdownarrows
          h \rightthreearrows h \leftthreearrows
          p \leftrightharpoons       p \rightleftharpoons
          p \updownharpoonsleftright p \downupharpoonsleftright p
        \]
  \item Math accents:
        \TopAccentMap{\symnormal}{x}
        % \begin{itemize}
        %   \item Latin capital letters:
        %     \TopAccentMap{\symnormal}{\LatinAlphabets}
        %   \item Latin small letters:
        %     \TopAccentMap{\symnormal}{\latinAlphabets}
        %   \item Latin capital upright letters:
        %     \TopAccentMap{\symup}{\LatinAlphabets}
        %   \item Latin small upright letters:
        %     \TopAccentMap{\symup}{\latinAlphabets}
        %   \item Latin capital bold letters:
        %     \TopAccentMap{\symbf}{\LatinAlphabets}
        %   \item Latin small bold letters:
        %     \TopAccentMap{\symbf}{\latinAlphabets}
        %   \item Latin capital bold upright letters:
        %     \TopAccentMap{\symbfup}{\LatinAlphabets}
        %   \item Latin small bold upright letters:
        %     \TopAccentMap{\symbfup}{\latinAlphabets}
        %   \item Greek capital letters:
        %     \TopAccentMap{\symnormal}{\GreekAlphabets}
        %   \item Greek small letters:
        %     \TopAccentMap{\symnormal}{\greekAlphabets}
        %   \item Greek capital upright letters:
        %     \TopAccentMap{\symup}{\GreekAlphabets}
        %   \item Greek small upright letters:
        %     \TopAccentMap{\symup}{\greekAlphabets}
        %   \item Greek capital bold letters:
        %     \TopAccentMap{\symbf}{\GreekAlphabets}
        %   \item Greek small bold letters:
        %     \TopAccentMap{\symbf}{\greekAlphabets}
        %   \item Greek capital bold upright letters:
        %     \TopAccentMap{\symbfup}{\GreekAlphabets}
        %   \item Greek small bold upright letters:
        %     \TopAccentMap{\symbfup}{\greekAlphabets}
        % \end{itemize}
  \item Integral:
        \[
            \int_0^\pi        \sin x \, \mathrm{d} x
          = \int\limits_0^\pi \sin x \, \mathrm{d} x
          = \cos 0 - \cos\pi + C
        \]
        \[
          \int_{-\infty}^{+\infty}    \mathrm{d} z
          \iint_{-\infty}^{+\infty}   \mathrm{d}^2 y
          \iiint_{-\infty}^{+\infty}  \mathrm{d}^3 x
          \iiiint_{-\infty}^{+\infty} \mathrm{d}^4 p
        \]
        \[ \oint \dd{r} \oiint \dd{\theta} \oiiint \dd{\varphi}\]
        \begingroup
          \mathversion{upintegral}
          \[
              \int_0^\pi        \sin x \, \mathrm{d} x
            = \int\limits_0^\pi \sin x \, \mathrm{d} x
            = \cos 0 - \cos\pi + C
          \]
          \[
            \int_{-\infty}^{+\infty}    \mathrm{d} z
            \iint_{-\infty}^{+\infty}   \mathrm{d}^2 y
            \iiint_{-\infty}^{+\infty}  \mathrm{d}^3 x
            \iiiint_{-\infty}^{+\infty} \mathrm{d}^4 p
          \]
          \[ \oint \dd{r} \oiint \dd{\theta} \oiiint \dd{\varphi} \]
        \endgroup
  \item Huge operators:
        \[
          \int\limits_0^\infty \int_0^\infty
          \sum_{i=1}^\infty \prod_{j=i}^\infty \coprod_{k=i}^\infty
        \]
        \[
          \sum_{i=1}^\infty    \frac{1}{x^i} = \frac{1}{1-x} \quad
          \prod_{i=1}^\infty   \frac{1}{x^i} = x^{-n(n+1)/2} \quad
          \coprod_{i=i}^\infty \frac{1}{x^i} = ?
        \]
  \item Huge operators (inline):
        $ \int\limits_0^\infty \int_0^\infty \iint \dd{x} \iiint \dd{y} \iiiint \dd{p}
          \oint \dd{r} \oiint \dd{\theta} \oiiint \dd{\varphi}
          \sum_{i=1}^\infty \prod_{j=i}^\infty \coprod_{i=i}^\infty $
  \item Huge operators (inline):
        \begingroup
          \mathversion{upintegral}
          $ \int\limits_0^\infty \int_0^\infty \iint \dd{x} \iiint \dd{y} \iiiint \dd{p}
            \oint \dd{r} \oiint \dd{\theta} \oiiint \dd{\varphi}
            \sum_{i=1}^\infty \prod_{j=i}^\infty \coprod_{i=i}^\infty $
        \endgroup
  \item Fraction:
        \[ \frac{1}{2} + \frac{1}{\frac{2}{3}+4} + \frac{\frac{1}{2}+3}{4} \]
  \item Fraction (inline):
        $ \frac{1}{2} + \frac{1g}{2} + \frac{1}{\frac{2}{3}+4} + \frac{\frac{1}{2}+3}{4} $
  \item Radical:
        \[
            \sqrt{2} + \sqrt{2^2} + \sqrt{1+\sqrt{2}} + \sqrt{1+\sqrt{1+\sqrt{3}}}
          + \sqrt{\sqrt{\sqrt{\sqrt{2}}}} + \sqrt{\frac{1}{2}}
        \]
        \[
            \cuberoot{2} + \cuberoot{2^2} + \cuberoot{1+\cuberoot{2}}
          + \cuberoot{1+\cuberoot{1+\cuberoot{3}}}
          + \cuberoot{\cuberoot{\cuberoot{\cuberoot{2}}}} + \cuberoot{\frac{1}{2}}
        \]
        \[
            \fourthroot{2} + \fourthroot{2^2} + \fourthroot{1+\fourthroot{2}}
          + \fourthroot{1+\fourthroot{1+\fourthroot{3}}}
          + \fourthroot{\fourthroot{\fourthroot{\fourthroot{2}}}} + \fourthroot{\frac{1}{2}}
        \]
        \[
            \sqrt[x]{y} + \sqrt[x]{\sqrt[x]{y}} + \sqrt[x]{\sqrt[x]{\sqrt[x]{y}}}
          + \sqrt[x]{\frac{1}{2}}
          + \sqrt      { \begin{matrix} x \\ y \\ z \\ w \end{matrix} }
          + \cuberoot  { \begin{matrix} x \\ y \\ z \\ w \end{matrix} }
          + \fourthroot{ \begin{matrix} x \\ y \\ z \\ w \end{matrix} }
          + \sqrt[x]   { \begin{matrix} x \\ y \\ z \\ w \end{matrix} }
          + \sqrt      { \begin{matrix} x \\ y \\ z \\ w \\ p \end{matrix} }
          + \cuberoot  { \begin{matrix} x \\ y \\ z \\ w \\ p \end{matrix} }
          + \fourthroot{ \begin{matrix} x \\ y \\ z \\ w \\ p \end{matrix} }
          + \sqrt[x]   { \begin{matrix} x \\ y \\ z \\ w \\ p \end{matrix} }
        \]
        \[ \PrintRadical{\sqrt}{x}{25}       \]
        \[ \PrintRadical{\cuberoot}{x}{25}   \]
        \[ \PrintRadical{\fourthroot}{x}{25} \]
        \[ \PrintRadical{\sqrt[x]}{x}{4}     \]
  \item Brackets:
        \[ (a) (A) (O) (Y) (y) (f) (Q) (T) (Y) (j) (q) \]
        % \[ \PrintDelimiters{(}{)}             \]
        % \[ \PrintDelimiters{\lgroup}{\rgroup} \]
        % \[ \PrintDelimiters{[}{]}             \]
        % \[ \PrintDelimiters{\{}{\}}           \]
        % \[ \PrintDelimiters{\vert}{\vert}     \]
        % \[ \PrintDelimiters{\Vert}{\Vert}     \]
        % \[ \PrintDelimiters{\Vvert}{\Vvert}   \]
        % \[ \PrintDelimiters{\langle}{\rangle} \]
        % \[ \PrintDelimiters{\lAngle}{\rAngle} \]
        % \[ \PrintDelimiters{\lceil}{\rceil}   \]
        % \[ \PrintDelimiters{\lfloor}{\rfloor} \]
        \[
          \Biggl(  \biggl(  \Bigl(  \bigl(   (x)  \bigr)  \Bigr)  \biggr)  \Biggr)  \quad
          \Biggl\lgroup \biggl\lgroup \Bigl\lgroup  \bigl\lgroup  \lgroup x \rgroup
          \bigr\rgroup  \Bigr\rgroup  \biggr\rgroup \Biggr\rgroup                   \quad
          \Biggl[  \biggl[  \Bigl[  \bigl[   [x]  \bigr]  \Bigr]  \biggr]  \Biggr]  \quad
          \Biggl\{ \biggl\{ \Bigl\{ \bigl\{ \{x\} \bigr\} \Bigr\} \biggr\} \Biggr\}
        \]
        \[
            \left( x \right) + \left( x^2 \right)
          + \left( \frac{1}{2} \right) + \left( \frac{2^2}{3} \right)
          + \left( \frac{\frac{1}{2}}{\frac{3}{4}} \right)
        \]
        \[
          ( \vert ) [ \Vert ] \{ \Vvert \} \quad
          \bigl(  \bigm\vert  \bigr)  \bigl[  \bigm\Vert  \bigr]  \bigl\{  \bigm\Vvert  \bigr\}  \quad
          \Bigl(  \Bigm\vert  \Bigr)  \Bigl[  \Bigm\Vert  \Bigr]  \Bigl\{  \Bigm\Vvert  \Bigr\}  \quad
          \biggl( \biggm\vert \biggr) \biggl[ \biggm\Vert \biggr] \biggl\{ \biggm\Vvert \biggr\} \quad
          \Biggl( \Biggm\vert \Biggr) \Biggl[ \Biggm\Vert \Biggr] \Biggl\{ \Biggm\Vvert \Biggr\} \quad
          \left(  \vbox to 40pt {} \middle\vert  \right)
          \left[  \vbox to 40pt {} \middle\Vert  \right]
          \left\{ \vbox to 40pt {} \middle\Vvert \right\} \quad
          \left(  \vbox to 50pt {} \middle\vert  \right)
          \left[  \vbox to 50pt {} \middle\Vert  \right]
          \left\{ \vbox to 50pt {} \middle\Vvert \right\}
        \]
  \item More brackets:
        \[
          \lceil  ceiling \rceil  \quad
          \lfloor floor   \rfloor \quad
          \lgroup group   \rgroup
        \]
  \item Bra-kets:
        \[
            \bra{x} + \ket{x} + \ip{\alpha}{\beta} + \op{\alpha^2}{\beta^2}
          + \bra{\frac{1}{2}} + \ket{\frac{1}{2}}
          + \ip{\frac{1}{2}}{\frac{1}{2}} + \op{\frac{1}{2}}{\frac{1}{2}}
          + \bra{\frac{a^2}{b^2}}
          + \Biggl\vert \frac{\mathrm{e}^{x^2}}{\mathrm{e}^{y^2}} \Biggr\rangle
        \]
        \[
            \langle \vert \rangle                   \quad
            \bigl\langle  \bigl\vert  \bigl\rangle  \quad
            \Bigl\langle  \Bigl\vert  \Bigl\rangle  \quad
            \biggl\langle \biggl\vert \biggl\rangle \quad
            \Biggl\langle \Biggl\vert \Biggl\rangle \qquad
            \lAngle \vert \rAngle                   \quad
            \bigl\lAngle  \bigl\vert  \bigl\rAngle  \quad
            \Bigl\lAngle  \Bigl\vert  \Bigl\rAngle  \quad
            \biggl\lAngle \biggl\vert \biggl\rAngle \quad
            \Biggl\lAngle \Biggl\vert \Biggl\rAngle
        \]
  \item Matrices:
        \[ \mqty(a & b \\ c & d) + \mqty*(a & b \\ c & d) \]
        \[
          \begin{pmatrix} \MatrixII  \end{pmatrix} \quad
          \begin{bmatrix} \MatrixII  \end{bmatrix} \quad
          \begin{Bmatrix} \MatrixII  \end{Bmatrix} \quad
          \begin{vmatrix} \MatrixII  \end{vmatrix} \quad
          \begin{Vmatrix} \MatrixII  \end{Vmatrix}
        \]
        \[
          \begin{pmatrix} \MatrixIII \end{pmatrix} \quad
          \begin{bmatrix} \MatrixIII \end{bmatrix} \quad
          \begin{Bmatrix} \MatrixIII \end{Bmatrix} \quad
          \begin{vmatrix} \MatrixIII \end{vmatrix} \quad
          \begin{Vmatrix} \MatrixIII \end{Vmatrix}
        \]
        \[
          \begin{pmatrix} \MatrixIV  \end{pmatrix} \quad
          \begin{bmatrix} \MatrixIV  \end{bmatrix} \quad
          \begin{Bmatrix} \MatrixIV  \end{Bmatrix} \quad
          \begin{vmatrix} \MatrixIV  \end{vmatrix} \quad
          \begin{Vmatrix} \MatrixIV  \end{Vmatrix}
        \]
  \item Nablas:
        \[ \nabla x + \grad{f} + \divergence{\symbf{u}} + \curl{\symbf{v}} \]
        \[
          \nabla                 \quad \symbf{\nabla}           \quad
          \symit{\nabla}         \quad \symbfit{\nabla};        \quad
          \tilde{\nabla}         \quad \tilde{\symbf{\nabla}}   \quad
          \tilde{\symit{\nabla}} \quad \tilde{\symbfit{\nabla}}
        \]
  \item Over-/underline and over-/underbraces
        \[ \OverUnderline{\overline}     \quad \overline     {\ListText}   \]
        \[ \OverUnderline{\overparen}    \quad \overparen    {\ListText}^n \]
        \[ \OverUnderline{\overbracket}  \quad \overbracket  {\ListText}^n \]
        \[ \OverUnderline{\overbrace}    \quad \overbrace    {\ListText}^n \]
        \[ \OverUnderline{\underline}    \quad \underline    {\ListText}   \]
        \[ \OverUnderline{\underparen}   \quad \underparen   {\ListText}_n \]
        \[ \OverUnderline{\underbracket} \quad \underbracket {\ListText}_n \]
        \[ \OverUnderline{\underbrace}   \quad \underbrace   {\ListText}_n \]
  \item Primes
        \[ x' x'' x''' x'''' x` x^{x'} x^{x''} x^{x'''} x^{x''''} x^{x`} \]
        \[ x \prime x \dprime x \trprime x \qprime \]
        \[ x^{\prime} x^{\dprime} x^{\trprime} x^{\qprime} \] % the same as ', '' or ''' => ssty
        \begin{center}
          \firatext x\symbol{"2032} x\symbol{"2033} x\symbol{"2034} x' x'' x'''
        \end{center}
\end{itemize}

\verb|\lim_{x\to\infty} \frac{1}{x^2} = 0|
\[ \lim_{x\to\infty} \frac{1}{x^2} = 0 \]

\verb|\frac{\partial y(x)}{\partial x} = \frac{\mathrm{d}y(x)}{\mathrm{d}x} = y'(x)|
\[ \frac{\partial y(x)}{\partial x} = \frac{\mathrm{d}y(x)}{\mathrm{d}x} = y'(x) \]



\iffalse
\subsection{More Samples}

\def\ee{\mathrm{e}}
\def\ii{\mathrm{i}}
\def\bm{\symbf}
\newcommand{\innerprod}[2]{\left\langle{#1}\middle\vert{#2}\right\rangle}
\newcommand{\brakket}[3]{\left\langle{#1}\middle\vert{#2}\middle\vert{#3}\right\rangle}
% \newcommand{\ket}[1]{\left\lvert{#1}\right\rangle}
% \newcommand{\bra}[1]{\left\langle{#1}\right\rvert}
% \newcommand{\ip}[2]{\left\langle{#1}\middle\vert{#2}\right\rangle}
% \newcommand{\op}[2]{\left\lvert{#1}\middle\rangle\middle\langle{#2}\right\rvert}
% \newcommand{\dd}{\,\mathrm{d}}
% \newcommand{\norm}[1]{\left\lVert{#1}\right\rVert}

\[ g^{mn} g_{mn} T^{i}_{jk} \]

\[ x \to \infty + \infty - \infty \]

\begin{align*}
     \int_{-\infty}^\infty \ee^{-x^2} \dd{x}
  &= \qty[\int_{-\infty}^\infty \ee^{-x^2} \dd{x} \, \int_{-\infty}^\infty \ee^{-y^2} \dd{y}]^{1/2} \\
  &= \qty[\int_0^{2\pi} \int_0^\infty \ee^{-r^2} r \dd{r}\dd{\theta}]^{1/2} \\
  &= \qty[\pi \int_0^\infty \ee^{-u} \dd{u}]^{1/2} \\
  &= \sqrt{\pi}
\end{align*}


\begin{align*}
\int_{0}^aJ_0\left[\frac{x_n^{(0)}}{a}r\right]J_0\left[\frac{x_m^{(0)}}{a}r\right]r\dd{r}=\frac{a^2}{2}J_1^2[x_n^{(0)}]\delta_m^n.\\
\int_{0}^{\infty}\frac{\cos x-\ee^{-x}}{x}\dd{x}=0\\
\end{align*}
\[\oint_{\partial\Sigma}\vec E\cdot \dd{\vec{l}}=-\frac{1}{c}\frac{\dd}{\dd t}\iint_{\Sigma}\vec B \cdot \dd{\vec{S}};\]
\[\partial_{[a}F_{\beta\gamma]}=0;\quad \partial_\alpha F^{\alpha\beta}=\mu_0J^\beta\]
\[\left(\frac{-\hbar^2}{2m}\nabla^2+V\right)\Psi=i\hbar\dot{\Psi}\]
\[\begin{split}
\frac{1}{\mathcal{C}^2}&{}=\frac{\innerprod{g'}{g'}}{\mathcal{C}^2}=\sum_{\bm{k}}\sum_{\bm{k}'}\brakket{g}{c_{\bm{k}',\uparrow}^\dagger c_{\bm{k}',\downarrow} c_{\bm{k},\downarrow}^\dagger c_{\bm{k},\uparrow}}{g}=\sum_{\bm{k}}\brakket{g}{c_{\bm{k},\uparrow}^\dagger c_{\bm{k},\downarrow} c_{\bm{k},\downarrow}^\dagger c_{\bm{k},\uparrow}}{g}\\
&{}=\sum_{\bm{k}}\brakket{g}{n_{\bm{k},\uparrow}\left(1-n_{\bm{k},\downarrow}\right)}{g}\\
&{}=\sum_{\norm{\bm{k}}<k_F^\downarrow}\brakket{g}{0}{g}+\sum_{k_F^\downarrow<\norm{\bm{k}}<k_F^\uparrow}\brakket{g}{1}{g}+\sum_{\norm{\bm{k}}>k_F^\uparrow}\brakket{g}{0}{g}\\
&{}=N_\uparrow-N_\downarrow
\end{split}\]
\[\left[ f,g \right]\equiv \sum_{\alpha =1}^{s}{\left( \frac{\partial f}{\partial {{q}_{\alpha }}}\frac{\partial g}{\partial {{p}_{\alpha }}}-\frac{\partial g}{\partial {{q}_{\alpha }}}\frac{\partial f}{\partial {{p}_{\alpha }}} \right)}=\sum\limits_{\alpha =1}^{s}{\begin{vmatrix}
	\partial_{{q}_{\alpha }} f & \partial_{{p}_{\alpha }} f  \\
	\partial_{{q}_{\alpha }} g & \partial_{{p}_{\alpha }} g  \\
	\end{vmatrix} }=\sum\limits_{\alpha =1}^{s}{\frac{\partial \left( f,g \right)}{\partial \left( {{q}_{\alpha }},{{p}_{\alpha }} \right)}}\]
\[\begin{split}
& \frac{{{\text{d}}^{2}}f}{\text{d}{{t}^{2}}}=\frac{\text{d}}{\text{d}t}\left[ f,H \right]=\left[ \left[ f,H \right],H \right]=\hat{H}\hat{H}f={{{\hat{H}}}^{2}}f \\ 
& \vdots  \\ 
& \frac{{{\text{d}}^{n}}f}{\text{d}{{t}^{n}}}=\underbrace{\left[ \left[ \left[ f,H \right],\cdots  \right],H \right]}_{n}={{{\hat{H}}}^{n}}f  \\
\end{split}\]
\[\tilde{U}(r,z)=E_0\dfrac{\omega_0}{\omega(z)}\exp\left[-r^2\left(\dfrac{1}{\omega^2(z)}+\dfrac{\ii k}{2R(z)}\right)-\ii k z+\ii \zeta(z)\right]\]
\[\omega(z)=\omega_0\sqrt{1+\left(\dfrac{\lambda z}{\pi {\omega_0}^2}\right)^2};\quad R(z)=z\left[1+\left(\dfrac{\pi {\omega_0}^2}{\lambda z}\right)^2\right]\]
\[\left( \begin{matrix}
{mg}/{l}\;+k-m\omega _{1}^{2} & -k  \\
-k & {mg}/{l}\;+k-m\omega _{1}^{2}  \\
\end{matrix} \right)\left( \begin{matrix}
{{a}_{11}}  \\
{{a}_{21}}  \\
\end{matrix} \right)=0\]
\[V=\underbrace{{{V}_{0}}}_{=0}+\underbrace{\sum\limits_{\alpha =1}^{s}{{{\left( \frac{\partial V}{\partial {{q}_{\alpha }}} \right)}_{0}}{{q}_{\alpha }}}}_{=0}+\underbrace{\frac{1}{2}\sum\limits_{\alpha ,\beta =1}^{s}{{{\left( \frac{{{\partial }^{2}}V}{\partial {{q}_{\alpha }}\partial {{q}_{\beta }}} \right)}_{0}}{{q}_{\alpha }}{{q}_{\beta }}}}_{>0}+\cdots \]
\[T=\frac{1}{2}\sum\limits_{i=1}^{n}{{{m}_{i}}{{{\dot{\bm r}}}_{i}}\cdot {{{\dot{\bm r}}}_{i}}}=\frac{1}{2}\sum\limits_{\alpha ,\beta =1}^{s}{\left[ \sum\limits_{i=1}^{n}{{{m}_{i}}{{\left( \frac{\partial {{\bm r}_{i}}}{\partial {{q}_{\alpha }}} \right)}_{0}}\cdot {{\left( \frac{\partial {{\bm r}_{i}}}{\partial {{q}_{\beta }}} \right)}_{0}}} \right]{{{\dot{q}}}_{\alpha }}{{{\dot{q}}}_{\beta }}}+\cdots \]
\[\left( \begin{matrix}
{{u}_{0}}  \\
{{u}_{1}}  \\
\vdots   \\
{{u}_{N-1}}  \\
\end{matrix} \right)=\sum\limits_{k>0}{\left[ \left( \begin{matrix}
	1  \\
	\cos ka  \\
	\vdots   \\
	\cos k\left( N-1 \right)a  \\
	\end{matrix} \right)\underbrace{{{C}_{k+}}\cos \left( {{\omega }_{k}}t+{{\varphi }_{k+}} \right)}_{\frac{2}{\sqrt{N}}{{q}_{k+}}}+\left( \begin{matrix}
	0  \\
	\sin ka  \\
	\vdots   \\
	\sin k\left( N-1 \right)a  \\
	\end{matrix} \right)\underbrace{{{C}_{k-}}\cos \left( {{\omega }_{k}}t+{{\varphi }_{k-}} \right)}_{\frac{2}{\sqrt{N}}{{q}_{k-}}} \right]}\]
\[G(\vec{r},{\vec{r}}',\tau )=\int _{-\infty }^{\infty }\tilde{G}(\vec{r},{\vec{r}}',\omega )e^{-i \tau  \omega }d\omega=\int_{-\infty }^{\infty } \frac{e^{-i \tau  \omega } e^{i k |\vec{r}-{\vec{r}}'| }}{(2 \pi ) |\vec{r}-{\vec{r}}'| } \, d\omega=\frac{\delta  \left(\tau -\frac{R}{c}\right)}{|\vec{r}-{\vec{r}}'| }\]
\[
\begin{split}	
\mathcal{F}^{-1}(\ket{j})
&{}=\frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1}\exp\left(-2\uppi \ii \frac{jk}{2^n}\right)\ket{k}.\\
&{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\exp\left(-2\uppi \ii j\sum_{l=0}^{n-1}\frac{2^l k_l}{2^n}\right)\ket{k_{n-1}\cdots k_0}\\
&{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\bigotimes_{l=1}^n\left[\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\ket{k_{n-l}}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\sum_{k_{n-l}=0}^1\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\ket{k_{n-l}}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\ket{0}_{n-l}+\ee^{-2\uppi \ii j /2^l}\ket{1}_{n-l}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\ket{0}_{n-l}+\ee^{-2\uppi \ii ({0.j_{l-1}\ldots j_0})}\ket{1}_{n-l}\right].
\end{split}
\]

\newcommand{\lb}{\left(}
\newcommand{\rb}{\right)}
\newcommand{\ec}{\text{,}}
\newcommand{\ed}{\text{.}}
\newcommand{\bt}{\lb t\rb}
\newcommand{\deltaup}{\updelta}
\newcommand{\piup}{\uppi}
\newcommand{\ndd}{\,\mathrm{d}}
\subsubsection*{Problem 1}
For convenience, first we set $t_i=0$, and in the end, we replace $t_f$ by $t_f-t_i$ and right answer is obtained.
The classical path is \[x_c\lb t\rb=A \cos\omega t+B\sin \omega t\ec\]where $A$ and $B$ can be determined by plugging $\lb0,x_i\rb$ and $\lb t_f, x_f\rb$ into the equation. The result is
\[x_c\lb t\rb=x_i \cos\omega t+\frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\sin \omega t\ed\]
We write $x\lb t\rb=x_c\lb t\rb+\deltaup x\bt$. Due to the fact that $\deltaup x$ should vanish at $t=0$ and $t=t_f$, $\deltaup x$ can be expanded as sine series: \[\deltaup x\bt=\sum_{n=1}^\infty a_n\sin\frac{n\piup t}{t_f}\ed\]
Also, the functional integral can be rewritten as \[\int\mathcal{D}\left[x\bt\right]=c\int\prod_{n=1}^\infty \dd a_n\ed\]
So, we have
\[L=\frac{m}{2}\lb\dot{x}_c+\deltaup\dot{x}\rb^2-\frac{m\omega^2}{2}\lb x_c+\deltaup x\rb^2\ec\]
\[\dot{x}\bt=-\omega x_i \sin\omega t +\omega \frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\cos\omega t+\sum_{n=1}^\infty\frac{a_n n \piup}{t_f}\cos\frac{n \piup t}{t_f}\ec\]
\[S=\int_0^{t_f} L\ndd t\ed\]
Because $x_c$ is the classical path, $\deltaup S_\text{classical}=0$, so there can't be any the linear term in the expression of $S$, and we also have in mind that the sine and cosine series are orthogonal. So, we can write S as following:
\[\begin{split}S&{}=\frac{m}{2}\int_0^{t_f}\left[\lb-\omega x_i\sin\omega t+\omega \frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\cos\omega t\rb^2+\sum_{n=1}^\infty\lb\frac{a_n n \piup}{t_f}\rb^2\cos^2\frac{n \piup t}{t_f}\right]\ndd t\\%
&\quad{}-\frac{m\omega^2}{2}\int_0^{t_f}\left[\lb x_i\cos\omega t+ \frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\sin\omega t\rb^2+\sum_{n=1}^\infty {a_n}^2\sin^2\frac{n \piup t}{t_f}\right]\ndd t\\%
&{}=\sum_{n=1}^\infty\int_0^{t_f}\left[\frac{m}{2}\lb\frac{a_n n \piup}{t_f}\rb^2\cos^2\frac{n \piup t}{t_f}-\frac{m\omega^2}{2}{a_n}^2\sin^2\frac{n \piup t}{t_f}\right]\ndd t\\%
&\quad{}+\frac{m\omega^2}{2}\int_0^{t_f}\left[ {x_i}^2-\lb\frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\rb^2\right]\lb\sin^2\omega t-\cos^2\omega t\rb\ndd t\\%
&\quad{}-\frac{m\omega^2}{2}\int_0^{t_f}4 {x_i}\lb\frac{x_f-x_i\cos\omega t_f}{\sin\omega t_f}\rb\lb\sin\omega t\cos\omega t\rb\ndd t\ed\end{split}\]
Using 
\[\int_0^{t_f}\lb\sin^2\omega t-\cos^2\omega t\rb\ndd t=-\frac{\sin2\omega t_f}{2\omega}\ec\]
\[\int_0^{t_f}\sin\omega t\cdot\cos\omega t\ndd t=\frac{\sin^2\omega t_f}{2\omega}\ec\]
\[\int_0^{t_f}\sin^2\frac{n\piup t}{t_f} \ndd t=\int_0^{t_f}\cos^2\frac{n\piup t}{t_f} \ndd t=\frac{a_n n \piup}{t_f}\ec\]
we get
\[S=\sum_{n=1}^\infty\left[\frac{m}{2}\lb\frac{a_n n \piup}{t_f}\rb^2-\frac{m\omega^2}{2}{a_n}^2\right]\frac{t_f}{2}+\frac{m\omega}{2\sin\omega t_f}\left[\lb {x_i}^2+{x_f}^2\rb\cos\omega t_f-2 x_i x_f\right]\ed\]
\[\begin{split}U={}&\exp\left\{\frac{\ii}{\hbar}\frac{m\omega}{2\sin\omega t_f}\left[\lb {x_i}^2+{x_f}^2\rb\cos\omega t_f-2 x_i x_f\right]\right\}\\%
&{}\times c\prod_{n=1}^{\infty}\int_{-\infty}^\infty\exp{\frac{\ii}{\hbar}\left[\frac{m}{2}\lb\frac{n \piup}{t_f}\rb^2-\frac{m\omega^2}{2}\right]\frac{t_f {a_n}^2}{2}}\ndd a_n\ed\end{split}\]
Using the Fresnel integral formula:
\[\int_{-\infty}^\infty\exp\lb \ii t\rb\ndd t=\sqrt{\piup \ii}\ec\]
\[\int_{-\infty}^\infty\exp{\frac{\ii}{\hbar}\left[\frac{m}{2}\lb\frac{n \piup}{t_f}\rb^2-\frac{m\omega^2}{2}\right]\frac{t_f {a_n}^2}{2}}\ndd a_n\sim\frac{\sqrt{t_f}}{n}\ec\]
\[U\lb x_f,t_f;x_i,t_i\rb=c'\lb t_f-t_i\rb\exp\left\{\frac{\ii m\omega}{2\hbar\sin\left[\omega \lb t_f-t_i\rb\right]}\left[\lb {x_i}^2+{x_f}^2\rb\cos\left[\omega\lb t_f-t_i\rb\right]-2 x_i x_f\right]\right\}\ed\]
Because \[\int\dd x U\lb x_f,t_f;x,t\rb U\lb x,t;x_i,t_i\rb=U\lb x_f,t_f;x_i,t_i\rb\ec\]
By using the Fresnel integral again:
\[c'\lb t_f-t\rb c'\lb t-t_i\rb\sqrt{\frac{2 \piup \ii \hbar}{m \omega}\lb\frac{\cos\left[\omega\lb t_f-t\rb\right]}{\sin\left[\omega\lb t_f-t\rb\right]}+\frac{\cos\left[\omega\lb t-t_i\rb\right]}{\sin\left[\omega\lb t-t_i\rb\right]}\rb}=c'\lb t_f-t_i\rb\ec\]
\[c'\lb t_f-t_i\rb=\sqrt{\frac{m\omega}{2\piup \ii \hbar\sin\left[\omega\lb t_f-t_i\rb\right]}}\ed\]
Thus
\[\begin{split}U\lb x_f,t_f;x_i,t_i\rb=&\sqrt{\frac{m\omega}{2\piup \ii \hbar\sin\left[\omega\lb t_f-t_i\rb\right]}}\\&{}\times\exp\left\{\frac{\ii m\omega}{2\hbar\sin\left[\omega \lb t_f-t_i\rb\right]}\left[\lb {x_i}^2+{x_f}^2\rb\cos\left[\omega\lb t_f-t_i\rb\right]-2 x_i x_f\right]\right\}\ed\end{split}\]


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