<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,fleqn]{article}
\usepackage[a4paper, margin=1in]{geometry}
\usepackage{amsmath}
\usepackage[math-style=ISO, bold-style=ISO]{unicode-math}
\usepackage{metalogo}
\usepackage{extarrows}
\makeatletter
\renewcommand{\relbar}{\symbol{"E010}\mkern-.2mu\symbol{"E010}\mkern1.8mu}
\renewcommand{\Relbar}{\symbol{"E011}\mkern-.2mu\symbol{"E011}\mkern1.8mu}
\makeatother

% \setmainfont{EB Garamond}
\setmainfont{EB Garamond}
% \setmonofont{Source Code Pro}[Scale=.86]
\setmathfont{Garamond-Math.otf}[Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathI   ,StylisticSet={1 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathII  ,StylisticSet={2 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathIII ,StylisticSet={3 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathIV  ,StylisticSet={4 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathV   ,StylisticSet={5 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathVI  ,StylisticSet={6 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathVII ,StylisticSet={7 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathVIII,StylisticSet={8 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathIX  ,StylisticSet={9 },Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathX   ,StylisticSet={10},Path=../Release/ ]%, Scale=MatchUppercase]
\setmathfont{Garamond-Math.otf}[version=GaramondMathXI   ,StylisticSet={11},Path=../Release/ ]%, Scale=MatchUppercase]
\newfontfamily\GaramondMath{Garamond-Math.otf}[Path=../Release/]

\def\Latinalphabets{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\def\latinalphabets{abcdefghijklmnopqrstuvwxyz}
\def\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
}
\def\greekalphabets{%
  \alpha      \beta       \gamma      \delta      \epsilon    \varepsilon
   \zeta       \eta        \theta      \vartheta 
  \iota       \kappa      \varkappa   \lambda     \mu
  \nu         \xi         \omicron    \pi \varpi        \rho
  \varrho     \sigma      \varsigma   \tau        \upsilon
  \phi        \varphi     \chi        \psi        \omega
}
% ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
% αβγδεζηθικλμνξοπρστυφχψω
\def\TOPACCENT#1{%
  \acute{#1}    \, \bar{#1}   \, \breve{#1} \, \check{#1}   \, \ddddot{#1}    , \quad
  \dddot{#1}    \, \ddot{#1}  \, \dot{#1}   \, \grave{#1}   \, \hat{#1}       , \quad
  \mathring{#1} \, \tilde{#1} \, \vec{#1}   \, \widehat{#1} \, \widetilde{#1}
}
\ExplSyntaxOn
\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} { \[ \TOPACCENT{#1{##1}} \] } }
\cs_generate_variant:Nn \fonttest_top_accent_map:Nn { Nx }
\NewDocumentCommand \SUPSUBMAP { m m }
  { \[ \exp_args:Nx \tl_map_inline:nn {#2} {{ #1{##1 \sb{\QED}} }} \] }
\NewDocumentCommand \CIRCLEDNUMA { s }
  {
    % with *:    0-50
    % without *: 0-10
    \symbol {"24EA}
    \fonttest_circled_aux:nn {"2460} {"2469}
    \IfBooleanT {#1}
      {
        \fonttest_circled_aux:nn {"246A} {"2473}
        \fonttest_circled_aux:nn {"3251} {"325F}
        \fonttest_circled_aux:nn {"32B1} {"32BF}
      }
  }
\NewDocumentCommand \CIRCLEDNUMB { s }
  {
    % with *:    0-20
    % without *: 0-10
    \symbol {"24FF}
    \fonttest_circled_aux:nn {"2776} {"277F}
    \IfBooleanT {#1}
      { \fonttest_circled_aux:nn {"24EB} {"24F4} }
  }
\NewDocumentCommand \CIRCLEDNUMC { }
  { \fonttest_circled_aux:nn {"24F5} {"24FE} }
\NewDocumentCommand \CIRCLEDLETTERA { }
  { \fonttest_circled_aux:nn {"24B6} {"24CF} }
\NewDocumentCommand \CIRCLEDLETTERB { }
  { \fonttest_circled_aux:nn {"1F150} {"1F169} }
\NewDocumentCommand \CIRCLEDLETTERC { }
  { \fonttest_circled_aux:nn {"24D0} {"24E9} }
\NewDocumentCommand \CHESSSYMB { }
  { \fonttest_circled_aux:nn {"2654} {"265F} }
\cs_new:Npn \fonttest_circled_aux:nn #1#2
  { \int_step_inline:nnn {#1} {#2} { \symbol {##1} } }
\ExplSyntaxOff
\def\OVERUNDERLINE#1{%
  #1{} \quad #1{b} \quad #1{ab} \quad #1{abc} \quad #1{abcd} \quad #1{abcde} \quad #1{a+b+c}}
\def\LISTTEXT{x_1, \, x_2, \, x_3,\ x_4\, \ \ldots, \, x_n}

\DeclareRobustCommand{\GenericInfo}[2]{}

\def\ee{\symrm{e}}
\def\ii{\symrm{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{\kets}[1]{\lvert{#1}\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}{\text{d}}
\newcommand{\norm}[1]{\left\lVert{#1}\right\rVert}

\title{Garamond-Math, Ver. 2022-01-03}
\author{Yuansheng Zhao, Xiangdong Zeng}

\begin{document}
\maketitle
\section{Introduction}
Garamond-Math is an open type math font matching the \emph{EB Garamond (Octavio Pardo)}\footnote{https://ctan.org/pkg/ebgaramond/, and https://github.com/octaviopardo/EBGaramond12/} and \emph{EB Garamond (Georg Mayr-Duffner)}\footnote{https://github.com/georgd/EB-Garamond/}. Many mathematical symbols are derived from other fonts, others are made from scratch. The metric is generated with a python script. Issues, bug reports, forks and other contributions are welcome. Please visit GitHub\footnote{https://github.com/YuanshengZhao/Garamond-Math/} for development details.

A minimal example with \texttt{unicode-math} package is as following:
\begin{verbatim}
    %Compile with `xelatex' command 
    \documentclass{article}
    \usepackage[math-style=ISO, bold-style=ISO]{unicode-math}
    \setmainfont{EB Garamond}%You should have installed the font
    \setmathfont{Garamond-Math.otf}[StylisticSet={7,9}]%Use StylisticSet that you like
    \begin{document}
        \[x^3+y^3=z^3.\]
    \end{document}
\end{verbatim}
The result should be
\[x^3+y^3=z^3.\]

\section{Alphabets \&amp; StylisticSets}
\subsubsection*{Latin and Greek (StylisticSet 4/5 give semi/extra bold for \texttt{\backslash symbf})}
\[ \Latinalphabets\]
\[ \latinalphabets \]
\[ \symup{\Latinalphabets}\]
\[ \symup{\latinalphabets} \]
\[ \symbf{\Latinalphabets}\]
\[ \symbf{\latinalphabets}\]
\[ \symbfup{\Latinalphabets}\]
\[  \symbfup{\latinalphabets} \]
\[ \Greekalphabets \]
\[\greekalphabets\]
\[ \symup{\Greekalphabets} \]
\[\symup{\greekalphabets} \]
\[ \symbf{\Greekalphabets} \]
\[\symbf{\greekalphabets} \]
\[ \symbfup{\Greekalphabets}\]
\[ \symbfup{\greekalphabets} \]
\begingroup\mathversion{GaramondMathIV}\[\symbf{\Latinalphabets}\]
\[\symbf{\latinalphabets}\]\endgroup
\begingroup\mathversion{GaramondMathV}\[\symbf{\Latinalphabets}\]
\[\symbf{\latinalphabets}\]\endgroup

\subsubsection*{Sans and Typewriter: From Libertinus Math\footnote{https://github.com/khaledhosny/libertinus/}}
\[ \symsf{\Latinalphabets} \]
\[\symsf{\latinalphabets} \]
\[ \symsfup{\Latinalphabets} \]
\[\symsfup{\latinalphabets} \]
\[ \symbfsf{\Latinalphabets} \]
\[\symbfsf{\latinalphabets} \]
\[ \symbfsfup{\Latinalphabets} \]
\[\symbfsfup{\latinalphabets} \]
\[ \symtt{\Latinalphabets}\]
\[\symtt{\latinalphabets} \]

\subsubsection*{Blackboard (StylisticSet 1 $\rightarrow$ rounded XITS Math\footnote{https://github.com/khaledhosny/xits/})}
\[ \symbb{\Latinalphabets} \]
\[\symbb{\latinalphabets} \]
\begingroup\mathversion{GaramondMathI}\[\symbb{\Latinalphabets}\]
\[\symbb{\latinalphabets}\]\endgroup

\subsubsection*{Script: Rounded XITS Math [StylisticSet 3 $\rightarrow$ scaled CM; 8 $\rightarrow$ Garamond-compatible ones (experimental)]}
\[ \symscr{\Latinalphabets} \]
\[\symscr{\latinalphabets} \]
\[ \symbfscr{\Latinalphabets} \]
\[\symbfscr{\latinalphabets} \]
\begingroup\mathversion{GaramondMathIII}\[\symscr{\Latinalphabets}\]
\[\symbfscr{\Latinalphabets}\]\endgroup
\begingroup\mathversion{GaramondMathVIII}\[\symscr{\Latinalphabets}\]
\[\symscr{\latinalphabets}\]\endgroup

\subsubsection*{Fraktur: From Noto Sans Math\footnote{https://github.com/googlefonts/noto-fonts/}}
\[ \symfrak{\Latinalphabets} \]
\[\symfrak{\latinalphabets} \]
\[ \symbffrak{\Latinalphabets} \]
\[\symbffrak{\latinalphabets} \]


\subsubsection*{Digits: Same width between weight and serif/sans}
\[3.141592653589793238462643383279502884197169399375105820974944592307816406286\]
\[\symsf{3.141592653589793238462643383279502884197169399375105820974944592307816406286}\]
\[\symbf{3.141592653589793238462643383279502884197169399375105820974944592307816406286}\]

\subsubsection*{\texttt{\backslash partial}: (StylisticSet 2 $\rightarrow$ curved ones)}
\[\partial_\mu(\symup\partial^\mu\phi)-\symbf{\epsilon^{\lambda\mu\nu}\partial_\mu(A_\lambda\symbfup\partial_\nu f)}\]
\begingroup\mathversion{GaramondMathII}\[\partial_\mu(\symup\partial^\mu\phi)-\symbf{\epsilon^{\lambda\mu\nu}\partial_\mu(A_\lambda\symbfup\partial_\nu f)}\]\endgroup

\subsubsection*{\texttt{\backslash hbar}: (StylisticSet 6 $\rightarrow$ horizontal bars)}
\[\text{$\hbar$\qquad \begingroup\mathversion{GaramondMathVI} $\hbar$\endgroup}\]

\subsubsection*{Italic $\symbf h$: (StylisticSet 10 $\rightarrow$ out-bending ones)}
\[\text{$\displaystyle\hbar=\frac {\symbf{h}}{2\uppi} $\qquad \begingroup\mathversion{GaramondMathX} $\displaystyle\hbar=\frac {\symbf{h}}{2\uppi} $\endgroup}\]

\subsubsection*{\texttt{\backslash tilde}: (StylisticSet 9 $\rightarrow$ ``normal'' ones)}
\[\text{$\tilde F$\qquad \begingroup\mathversion{GaramondMathIX} $\tilde F$\endgroup}\]

\subsubsection*{\texttt{\backslash int}: (StylisticSet 7 $\rightarrow$ a variant with inversion symmetry)}
\[\oint_{\partial\Sigma}\vec E\cdot \dd{\vec{l}}=-\frac{1}{c}\frac{\dd}{\dd t}\iint_{\Sigma}\vec B \cdot \dd{\vec{S}}\]
\begingroup\mathversion{GaramondMathVII}\[\oint_{\partial\Sigma}\vec E\cdot \dd{\vec{l}}=-\frac{1}{c}\frac{\dd}{\dd t}\iint_{\Sigma}\vec B \cdot \dd{\vec{S}}\]\endgroup

\subsubsection*{Binany Operators: (StylisticSet 11 $\rightarrow$ larger ones)}
\[s=A+b\times 1\div x^3\]
\begingroup\mathversion{GaramondMathXI}\[s=A+b\times 1\div x^3\]\endgroup

\subsubsection*{Other Symbols}
\begingroup
  \hspace{\parindent}\GaramondMath
  \CIRCLEDNUMA*   \par
  \CIRCLEDNUMB*   \par
  \CIRCLEDNUMC    \par
  \CIRCLEDLETTERA \par
  \CIRCLEDLETTERB \par
  \CIRCLEDLETTERC \par
  \CHESSSYMB
\endgroup

\subsubsection*{Extensible Arrow Hack}
The font contains the math table for constructing extensible arrow. However \texttt{unicode-math} does not provide an interface to that. In \LuaTeX ~one can use \texttt{\textbackslash Uhextensible}\footnote{https://tex.stackexchange.com/questions/423893/}. A more general solution is to add the following code in preamble.
\begin{verbatim}
\usepackage{extarrow} %or mathtools
\makeatletter
\renewcommand{\relbar}{\symbol{"E010}\mkern-.2mu\symbol{"E010}\mkern1.8mu}
\renewcommand{\Relbar}{\symbol{"E011}\mkern-.2mu\symbol{"E011}\mkern1.8mu}
\makeatother
\end{verbatim}
Then \texttt{\textbackslash xleftarrow} and other commands will work:
\[\mathrm{CH}_3\mathrm{COO}\mathrm{H}+\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\xrightarrow[{\triangle}]{\mathrm{H}_2\mathrm{SO}_4}\mathrm{CH}_3\mathrm{COOC}_2\mathrm{H}_5+\mathrm{H}_2\mathrm{O}.\]
 
\section{Known Issue}
\begin{itemize}
    \item Fake optical size. EB Garamond does not contain a complete set of glyphs (normal + bold + optical size of both weights). The ``optical size \texttt{ssty}'' is made by interpolating different weights at the present (without this, the double script is too thin to be readable). 
\end{itemize}

\section{Equation Samples}
\[ 1 + 2 - 3 \times 4 \div 5 \pm 6 \mp 7 \dotplus 8 = -a \oplus b \otimes c -\{z\}\]
\[\forall \epsilon, \exists \delta : x \in A \cup B \subset S \cap T \ntrianglerighteq U\] 
\[R_{\nu\kappa\lambda}^\mu=\partial_\kappa\Gamma_{\lambda\nu}^\mu-\partial_\lambda\Gamma_{\kappa\nu}^\mu+\Gamma_{\kappa\sigma}^\mu\Gamma_{\lambda\nu}^\sigma-\Gamma_{\lambda\sigma}^\mu\Gamma_{\kappa\nu}^\sigma\]
\[T_{\alpha_1\cdots\alpha_k}'^{\beta_1\cdots\beta_l}=T_{i_1\cdots i_k}^{j_1\cdots j_l}
    \frac{\partial x^{i_1}}{\partial x'^{\alpha_1}}\cdots
    \frac{\partial x^{i_k}}{\partial x'^{\alpha_k}}
    \frac{\partial x'^{\beta_1}}{\partial x^{j_1}}\cdots
    \frac{\partial x'^{\beta_l}}{\partial x^{j_l}}
\]
\[\int_{\sqrt{\frac{1-m u+m\Delta/k^2}{2mu/k}}}^{X_p}\widehat{1+2+3+4}+\widetilde{5+6+7+8}\]
\[ x \leftarrow y \leftrightarrow w \Rightarrow b \Leftrightarrow c
    \uparrow   y  \updownarrow    w \Downarrow  b \Updownarrow    c
    \Searrow p \Swarrow p
    x \leftharpoonup    x \upharpoonleft  
    X \mapsfrom Y \mapsto Z \mapsup 
    f \rightleftarrows  f \updownarrows f 
    h \rightthreearrows h \leftthreearrows
    p
\]
\[\int_0^1\frac{\ln (x+1)}{x}\dd{x}=\int_0^1\sum_{i=1}^{\infty}\frac{(-x)^{i-1}}{i}\dd{x}=\sum_{i=1}^{\infty}\int_0^1\frac{(-x)^{i-1}}{i}\dd{x}=\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i^2}=\frac{\uppi^2}{12}\]
\[
  \int\limits_0^\infty \int_0^\infty
  \sum_{i=1}^\infty \prod_{j=i}^\infty \coprod_{k=i}^\infty
  \oiiint \varointclockwise \ointctrclockwise \awint \intclockwise
\]
\[
  \Biggl(  \biggl(  \Bigl(  \bigl(   (x)  \bigr)  \Bigr)  \biggr)  \Biggr)  \quad
  \Biggl[  \biggl[  \Bigl[  \bigl[   [x]  \bigr]  \Bigr]  \biggr]  \Biggr]  \quad
  \Biggl\{ \biggl\{ \Bigl\{ \bigl\{ \{x\} \bigr\} \Bigr\} \biggr\} \Biggr\}\quad
   \Biggl\lvert \biggl\lvert \Bigl\lvert \bigl\lvert \lvert x\rvert \bigr\rvert \Bigr\rvert\biggr\rvert \Biggr\rvert\quad
   \Biggl\lVert \biggl\lVert \Bigl\lVert \bigl\lVert \lVert x\rVert \bigr\rVert \Bigr\rVert\biggr\rVert \Biggr\rVert\quad
    \Biggl\langle \biggl\langle \Bigl\langle \bigl\langle \langle x\rangle \bigr\rangle \Bigr\rangle\biggr\rangle \Biggr\rangle\quad
\]
\[       
 \Biggl\lgroup \biggl\lgroup \Bigl\lgroup \bigl\lgroup \lgroup x\rgroup \bigr\rgroup \Bigr\rgroup\biggr\rgroup \Biggr\rgroup\quad
\Biggl\lfloor \biggl\lfloor \Bigl\lfloor \bigl\lfloor \lfloor x\rfloor \bigr\rfloor \Bigr\rfloor\biggr\rfloor \Biggr\rfloor\quad
\Biggl\lceil \biggl\lceil \Bigl\lceil \bigl\lceil \lceil x\rceil \bigr\rceil \Bigr\rceil\biggr\rceil \Biggr\rceil\quad\]
\[
    \bra{x} + \ket{x} + \ip{\alpha}{\beta} + \op{\alpha}{\beta}
  + \bra{\frac{1}{2}} + \ket{\frac{1}{2}}
  + \ip{\frac{1}{2}}{\frac{1}{2}} + \op{\frac{1}{2}}{\frac{1}{2}}
  + \bra{\frac{a^2}{b^2}}
  + \Biggl\vert \frac{\ee^{x^2}}{\ee^{y^2}} \Biggr\rangle
\]
\[
  \CIRCLEDNUMB +
  ABC^{\CIRCLEDNUMA}
\]
\[\left( \begin{matrix}
{{u}_{0}}  \\
{{u}_{1}}  \\
\vdots   \\
{{u}_{N-1}}  \\
\end{matrix} \right)=\sum\limits_{k&gt;0}{\left[ \left( \begin{matrix}
    1  \\
    \cos ka  \\
    \vdots   \\
    \cos k\left( N-1 \right)a  \\
    \end{matrix} \right)\underbrace{{{C}_{k+}}\cos ( {{\omega }_{k}}t+{{\varphi }_{k+}} )}_{\frac{2}{\sqrt{N}}{{q}_{k+}}}+\left( \begin{matrix}
    0  \\
    \sin ka  \\
    \vdots   \\
    \sin k\left( N-1 \right)a  \\
    \end{matrix} \right)\underbrace{{{C}_{k-}}\cos ( {{\omega }_{k}}t+{{\varphi }_{k-}} )}_{\frac{2}{\sqrt{N}}{{q}_{k-}}} \right]}\]
\[
\begin{split}   
\mathcal{F}^{-1}(\kets{j})
&amp;{}=\frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1}\exp\left(-2\uppi \ii \frac{jk}{2^n}\right)\kets{k}.\\
&amp;{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\exp\left(-2\uppi \ii j\sum_{l=0}^{n-1}\frac{2^l k_l}{2^n}\right)\kets{k_{n-1}\cdots k_0}\\
&amp;{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\bigotimes_{l=1}^n\left[\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\kets{k_{n-l}}\right]\\
&amp;{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\sum_{k_{n-l}=0}^1\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\kets{k_{n-l}}\right]\\
&amp;{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\kets{0}_{n-l}+\ee^{-2\uppi \ii j /2^l}\kets{1}_{n-l}\right]\\
&amp;{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\kets{0}_{n-l}+\ee^{-2\uppi \ii (\overline{0.j_{l-1}\ldots j_0})}\kets{1}_{n-l}\right].
\end{split}
\]

\newcommand{\lb}{\left(}
\newcommand{\rb}{\right)}
\newcommand{\piup}{\uppi}
\newcommand{\ndd}{\,\mathrm{d}}
\[\begin{split}S&amp;{}=\frac{m}{2}\int_0^{t_{\text f}}\left[\lb-\omega x_{\text i}\sin\omega t+\omega \frac{x_{\text f}-x_{\text i}\cos\omega t_{\text f}}{\sin\omega t_{\text f}}\cos\omega t\rb^2+\sum_{n=1}^\infty\lb\frac{a_n n \piup}{t_{\text f}}\rb^2\cos^2\frac{n \piup t}{t_{\text f}}\right]\ndd t\\%
    &amp;\quad{}-\frac{m\omega^2}{2}\int_0^{t_{\text f}}\left[\lb x_{\text i}\cos\omega t+ \frac{x_{\text f}-x_{\text i}\cos\omega t_{\text f}}{\sin\omega t_{\text f}}\sin\omega t\rb^2+\sum_{n=1}^\infty {a_n}^2\sin^2\frac{n \piup t}{t_{\text f}}\right]\ndd t\\%
    &amp;{}=\sum_{n=1}^\infty\int_0^{t_{\text f}}\left[\frac{m}{2}\lb\frac{a_n n \piup}{t_{\text f}}\rb^2\cos^2\frac{n \piup t}{t_{\text f}}-\frac{m\omega^2}{2}{a_n}^2\sin^2\frac{n \piup t}{t_{\text f}}\right]\ndd t\\%
    &amp;\quad{}+\frac{m\omega^2}{2}\int_0^{t_{\text f}}\left[ {x_{\text i}}^2-\lb\frac{x_{\text f}-x_{\text i}\cos\omega t_{\text f}}{\sin\omega t_{\text f}}\rb^2\right]\lb\sin^2\omega t-\cos^2\omega t\rb\ndd t\\%
    &amp;\quad{}-\frac{m\omega^2}{2}\int_0^{t_{\text f}}4 {x_{\text i}}\lb\frac{x_{\text f}-x_{\text i}\cos\omega t_{\text f}}{\sin\omega t_{\text f}}\rb\lb\sin\omega t\cos\omega t\rb\ndd t.\end{split}\]
    \[\begin{split}U\lb x_{\text f},t_{\text f};x_{\text i},t_{\text i}\rb=&amp;\sqrt{\frac{m\omega}{2\piup \ii \hbar\sin\left[\omega\lb t_{\text f}-t_{\text i}\rb\right]}}\\&amp;{}\times\exp\left\{\frac{\ii m\omega}{2\hbar\sin\left[\omega \lb t_{\text f}-t_{\text i}\rb\right]}\left[\lb {x_{\text i}}^2+{x_{\text f}}^2\rb\cos\left[\omega\lb t_{\text f}-t_{\text i}\rb\right]-2 x_{\text i} x_{\text f}\right]\right\}.\end{split}\]
\end{document}


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