path: root/fonts/garamond-math/Garamond-Math.tex

\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]

\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} }
\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. 2019-08-16}
\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.

The font is mostly tested with \XeTeX, though it shoule also work with \LuaTeX.

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 shoule be
\[x^3+y^3=z^3.\]

\section{Alphabets \& 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 Typerwriter: 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*{Extensible Arrow Hack}
The font contains the math table for constructing extensible arrow. However \texttt{unicode-math} does not privode 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 Various spacing problems. Though math fonts technically should not be kerned, some pairs looks very ugly (Ex. $VA$); sometimes sub/superscript may also have same problem. However, do note that due to the mechanism in math mode, making all spacing look perfect is amlost impossible (as far as I can do, and low x-height and large italic angle only make things even worse), in many cases, adjusting manually (i.e. using \texttt{\textbackslash,} or \texttt{\textbackslash!}) is required.
    \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}
\[ 1 + 2 - 3 \times 4 \div 5 \pm 6 \mp 7 \dotplus 8 = -a \oplus b \otimes c -\{z\}\]
\[\forall \epsilon, \exists \delta : x \in A \cup B \subset S \cap T \ntrianglerighteq U\] 
\[R_{\nu\kappa\lambda}^\mu=\partial_\kappa\Gamma_{\lambda\nu}^\mu-\partial_\lambda\Gamma_{\kappa\nu}^\mu+\Gamma_{\kappa\sigma}^\mu\Gamma_{\lambda\nu}^\sigma-\Gamma_{\lambda\sigma}^\mu\Gamma_{\kappa\nu}^\sigma\]
\[T_{\alpha_1\cdots\alpha_k}'^{\beta_1\cdots\beta_l}=T_{i_1\cdots i_k}^{j_1\cdots j_l}
    \frac{\partial x^{i_1}}{\partial x'^{\alpha_1}}\cdots
    \frac{\partial x^{i_k}}{\partial x'^{\alpha_k}}
    \frac{\partial x'^{\beta_1}}{\partial x^{j_1}}\cdots
    \frac{\partial x'^{\beta_l}}{\partial x^{j_l}}
\]
\[\int_{\sqrt{\frac{1-m u+m\Delta/k^2}{2mu/k}}}^{X_p}\widehat{1+2+3+4}+\widetilde{5+6+7+8}\]
\[ x \leftarrow y \leftrightarrow w \Rightarrow b \Leftrightarrow c
    \uparrow   y  \updownarrow    w \Downarrow  b \Updownarrow    c
    \Searrow p \Swarrow p
    x \leftharpoonup    x \upharpoonleft  
    X \mapsfrom Y \mapsto Z \mapsup 
    f \rightleftarrows  f \updownarrows f 
    h \rightthreearrows h \leftthreearrows
    p
\]
\[\int_0^1\frac{\ln (x+1)}{x}\dd{x}=\int_0^1\sum_{i=1}^{\infty}\frac{(-x)^{i-1}}{i}\dd{x}=\sum_{i=1}^{\infty}\int_0^1\frac{(-x)^{i-1}}{i}\dd{x}=\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i^2}=\frac{\uppi^2}{12}\]
\[
  \int\limits_0^\infty \int_0^\infty
  \sum_{i=1}^\infty \prod_{j=i}^\infty \coprod_{k=i}^\infty
  \oiiint \varointclockwise \ointctrclockwise \awint \intclockwise
\]
\[
  \Biggl(  \biggl(  \Bigl(  \bigl(   (x)  \bigr)  \Bigr)  \biggr)  \Biggr)  \quad
  \Biggl[  \biggl[  \Bigl[  \bigl[   [x]  \bigr]  \Bigr]  \biggr]  \Biggr]  \quad
  \Biggl\{ \biggl\{ \Bigl\{ \bigl\{ \{x\} \bigr\} \Bigr\} \biggr\} \Biggr\}\quad
   \Biggl\lvert \biggl\lvert \Bigl\lvert \bigl\lvert \lvert x\rvert \bigr\rvert \Bigr\rvert\biggr\rvert \Biggr\rvert\quad
   \Biggl\lVert \biggl\lVert \Bigl\lVert \bigl\lVert \lVert x\rVert \bigr\rVert \Bigr\rVert\biggr\rVert \Biggr\rVert\quad
    \Biggl\langle \biggl\langle \Bigl\langle \bigl\langle \langle x\rangle \bigr\rangle \Bigr\rangle\biggr\rangle \Biggr\rangle\quad
\]
\[       
 \Biggl\lgroup \biggl\lgroup \Bigl\lgroup \bigl\lgroup \lgroup x\rgroup \bigr\rgroup \Bigr\rgroup\biggr\rgroup \Biggr\rgroup\quad
\Biggl\lfloor \biggl\lfloor \Bigl\lfloor \bigl\lfloor \lfloor x\rfloor \bigr\rfloor \Bigr\rfloor\biggr\rfloor \Biggr\rfloor\quad
\Biggl\lceil \biggl\lceil \Bigl\lceil \bigl\lceil \lceil x\rceil \bigr\rceil \Bigr\rceil\biggr\rceil \Biggr\rceil\quad\]
\[
    \bra{x} + \ket{x} + \ip{\alpha}{\beta} + \op{\alpha}{\beta}
  + \bra{\frac{1}{2}} + \ket{\frac{1}{2}}
  + \ip{\frac{1}{2}}{\frac{1}{2}} + \op{\frac{1}{2}}{\frac{1}{2}}
  + \bra{\frac{a^2}{b^2}}
  + \Biggl\vert \frac{\ee^{x^2}}{\ee^{y^2}} \Biggr\rangle
\]
\[
  \CIRCLEDNUMB +
  ABC^{\CIRCLEDNUMA}
\]
\[\left( \begin{matrix}
{{u}_{0}}  \\
{{u}_{1}}  \\
\vdots   \\
{{u}_{N-1}}  \\
\end{matrix} \right)=\sum\limits_{k>0}{\left[ \left( \begin{matrix}
    1  \\
    \cos ka  \\
    \vdots   \\
    \cos k\left( N-1 \right)a  \\
    \end{matrix} \right)\underbrace{{{C}_{k+}}\cos ( {{\omega }_{k}}t+{{\varphi }_{k+}} )}_{\frac{2}{\sqrt{N}}{{q}_{k+}}}+\left( \begin{matrix}
    0  \\
    \sin ka  \\
    \vdots   \\
    \sin k\left( N-1 \right)a  \\
    \end{matrix} \right)\underbrace{{{C}_{k-}}\cos ( {{\omega }_{k}}t+{{\varphi }_{k-}} )}_{\frac{2}{\sqrt{N}}{{q}_{k-}}} \right]}\]
\[
\begin{split}   
\mathcal{F}^{-1}(\kets{j})
&{}=\frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1}\exp\left(-2\uppi \ii \frac{jk}{2^n}\right)\kets{k}.\\
&{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\exp\left(-2\uppi \ii j\sum_{l=0}^{n-1}\frac{2^l k_l}{2^n}\right)\kets{k_{n-1}\cdots k_0}\\
&{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\bigotimes_{l=1}^n\left[\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\kets{k_{n-l}}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\sum_{k_{n-l}=0}^1\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\kets{k_{n-l}}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\kets{0}_{n-l}+\ee^{-2\uppi \ii j /2^l}\kets{1}_{n-l}\right]\\
&{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\kets{0}_{n-l}+\ee^{-2\uppi \ii (\overline{0.j_{l-1}\ldots j_0})}\kets{1}_{n-l}\right].
\end{split}
\]

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