%% $Id: firamath-otf-doc.tex 1022 2019-04-03 12:00:29Z herbert $ \listfiles \documentclass[english,log-declarations=false]{article} \usepackage{amsmath,esvect} \usepackage{FiraSans} \setmonofont{FiraMono}[ Numbers = {Monospaced}, Scale=MatchUppercase,FakeStretch=0.93] \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} \DeclareMathOperator{\Div}{\symup{div}} \DeclareMathOperator{\Grad}{\symup{grad}} \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[]{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{\symbf{\symup{T}}} \def\DD{\symbf{\symup{D}}} \subsection{Version normal} \begin{align} \begin{aligned} \Q{\varrho}{t}+\Div(\varrho\vec{v}) &= 0 \\ \varrho\Q{\vec{v}}{t}+(\varrho\vec{v}\cdot\nabla)\vec{v} &= \vec{f}_0+\Div\TT=\vec{f}_0 -\Grad p+\Div\TT' \\ \varrho T\frac{\diff s}{\diff t} &= \varrho\frac{\diff e}{\diff t} -\frac{p}{\varrho}\frac{\diff\varrho}{\diff t}=-\Div\vec{q}+\TT':\DD \end{aligned} \end{align} \begin{align} \Q{}{t}\iiint\<\varrho\diff^3V+\oiint\varrho(\vec{v}\cdot\vec{v}\vec{n})\diff^2A &= 0\\ \Q{}{t}\iiint\<\varrho\vec{v}\diff^3V+\oiint\varrho\vec{v}(\vec{v}\cdot\vec{n}\,)\diff^2A &= \iiint\ 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: \[ \int_0^\pi \sin x \, \mathrm{d} x = \int\limits_0^\pi \sin x \, \mathrm{d} x = \cos 0 - \cos\pi=2 \] \[ \int_{-\infty}^{+\infty} \mathrm{d} z \iint_{-\infty}^{+\infty} \mathrm{d}^2 y \iiint_{-\infty}^{+\infty} \mathrm{d}^3 x \iiiint_{-\infty}^{+\infty} \mathrm{d}^4 p \] \[ \oint \dd{r} \oiint \dd{\theta} \oiiint \dd{\varphi}\] \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:\\ \renewcommand\ket[1]{\left\lvert{#1}\right\rangle} \renewcommand\bra[1]{\left\langle{#1}\right\rvert} \renewcommand\ip[2]{\left\langle{#1}\middle\vert{#2}\right\rangle} \renewcommand\op[2]{\left\lvert{#1}\middle\rangle\middle\langle{#2}\right\rvert} \[ \bra{x} + \ket{x} + \ip{\alpha}{\beta} + \op{\alpha^2}{\beta^2} + \bra{\frac{1}{2}} + \ket{\frac{1}{2}} + \ip{\frac{1}{2}}{\frac{1}{2}} + \op{\frac{1}{2}}{\frac{1}{2}} + \bra{\frac{a^2}{b^2}} + \Biggl\vert \frac{\mathrm{e}^{x^2}}{\mathrm{e}^{y^2}} \Biggr\rangle \] \[ \langle \vert \rangle \quad \bigl\langle \bigl\vert \bigl\rangle \quad \Bigl\langle \Bigl\vert \Bigl\rangle \quad \biggl\langle \biggl\vert \biggl\rangle \quad \Biggl\langle \Biggl\vert \Biggl\rangle \qquad \lAngle \vert \rAngle \quad \bigl\lAngle \bigl\vert \bigl\rAngle \quad \Bigl\lAngle \Bigl\vert \Bigl\rAngle \quad \biggl\lAngle \biggl\vert \biggl\rAngle \quad \Biggl\lAngle \Biggl\vert \Biggl\rAngle \] \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 \[ x' x'' x''' x'''' x^{x'} x^{x''} x^{x'''} x^{x''''} x^{x`} \] \[ x \prime x \dprime x \trprime x \qprime \] \[ x^{\prime} x^{\dprime} x^{\trprime} x^{\qprime} \] % 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\limits_{x\to\infty} \frac{1}{x^2} = 0| $ \lim\limits_{x\to\infty} \dfrac{1}{x^2} = 0 $ \verb|\frac{\partial y(x)}{\partial x} = \frac{\symup{d}y(x)}{\symup{d}x} = y'(x)| \[ \frac{\partial y(x)}{\partial x} = \frac{\symup{d}y(x)}{\symup{d}x} = y'(x) \] \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{\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*} \[\oint_{\partial\Sigma}\vec E\cdot \dd{\vec{l}}=-\frac{1}{c}\frac{\dd}{\dd t}\iint_{\Sigma}\vec B \cdot \dd{\vec{S}};\] \[\partial_{[a}F_{\beta\gamma]}=0;\quad \partial_\alpha F^{\alpha\beta}=\mu_0J^\beta\] \[\left(\frac{-\hbar^2}{2m}\nabla^2+V\right)\Psi=i\hbar\dot{\Psi}\] \[\begin{split} \frac{1}{\mathcal{C}^2}&{}=\frac{\innerprod{g'}{g'}}{\mathcal{C}^2}=\sum_{\bm{k}}\sum_{\bm{k}'}\brakket{g}{c_{\bm{k}',\uparrow}^\dagger c_{\bm{k}',\downarrow} c_{\bm{k},\downarrow}^\dagger c_{\bm{k},\uparrow}}{g}=\sum_{\bm{k}}\brakket{g}{c_{\bm{k},\uparrow}^\dagger c_{\bm{k},\downarrow} c_{\bm{k},\downarrow}^\dagger c_{\bm{k},\uparrow}}{g}\\ &{}=\sum_{\bm{k}}\brakket{g}{n_{\bm{k},\uparrow}\left(1-n_{\bm{k},\downarrow}\right)}{g}\\ &{}=\sum_{\norm{\bm{k}}k_F^\uparrow}\brakket{g}{0}{g}\\ &{}=N_\uparrow-N_\downarrow \end{split}\] \[\left[ f,g \right]\equiv \sum_{\alpha =1}^{s}{\left( \frac{\partial f}{\partial {{q}_{\alpha }}}\frac{\partial g}{\partial {{p}_{\alpha }}}-\frac{\partial g}{\partial {{q}_{\alpha }}}\frac{\partial f}{\partial {{p}_{\alpha }}} \right)}=\sum\limits_{\alpha =1}^{s}{\begin{vmatrix} \partial_{{q}_{\alpha }} f & \partial_{{p}_{\alpha }} f \\ \partial_{{q}_{\alpha }} g & \partial_{{p}_{\alpha }} g \\ \end{vmatrix} }=\sum\limits_{\alpha =1}^{s}{\frac{\partial \left( f,g \right)}{\partial \left( {{q}_{\alpha }},{{p}_{\alpha }} \right)}}\] \[\begin{split} & \frac{{{\text{d}}^{2}}f}{\text{d}{{t}^{2}}}=\frac{\text{d}}{\text{d}t}\left[ f,H \right]=\left[ \left[ f,H \right],H \right]=\hat{H}\hat{H}f={{{\hat{H}}}^{2}}f \\ & \vdots \\ & \frac{{{\text{d}}^{n}}f}{\text{d}{{t}^{n}}}=\underbrace{\left[ \left[ \left[ f,H \right],\cdots \right],H \right]}_{n}={{{\hat{H}}}^{n}}f \\ \end{split}\] \[\tilde{U}(r,z)=E_0\dfrac{\omega_0}{\omega(z)}\exp\left[-r^2\left(\dfrac{1}{\omega^2(z)}+\dfrac{\ii k}{2R(z)}\right)-\ii k z+\ii \zeta(z)\right]\] \[\omega(z)=\omega_0\sqrt{1+\left(\dfrac{\lambda z}{\pi {\omega_0}^2}\right)^2};\quad R(z)=z\left[1+\left(\dfrac{\pi {\omega_0}^2}{\lambda z}\right)^2\right]\] \[\left( \begin{matrix} {mg}/{l}\;+k-m\omega _{1}^{2} & -k \\ -k & {mg}/{l}\;+k-m\omega _{1}^{2} \\ \end{matrix} \right)\left( \begin{matrix} {{a}_{11}} \\ {{a}_{21}} \\ \end{matrix} \right)=0\] \[V=\underbrace{{{V}_{0}}}_{=0}+\underbrace{\sum\limits_{\alpha =1}^{s}{{{\left( \frac{\partial V}{\partial {{q}_{\alpha }}} \right)}_{0}}{{q}_{\alpha }}}}_{=0}+\underbrace{\frac{1}{2}\sum\limits_{\alpha ,\beta =1}^{s}{{{\left( \frac{{{\partial }^{2}}V}{\partial {{q}_{\alpha }}\partial {{q}_{\beta }}} \right)}_{0}}{{q}_{\alpha }}{{q}_{\beta }}}}_{>0}+\cdots \] \[T=\frac{1}{2}\sum\limits_{i=1}^{n}{{{m}_{i}}{{{\dot{\bm r}}}_{i}}\cdot {{{\dot{\bm r}}}_{i}}}=\frac{1}{2}\sum\limits_{\alpha ,\beta =1}^{s}{\left[ \sum\limits_{i=1}^{n}{{{m}_{i}}{{\left( \frac{\partial {{\bm r}_{i}}}{\partial {{q}_{\alpha }}} \right)}_{0}}\cdot {{\left( \frac{\partial {{\bm r}_{i}}}{\partial {{q}_{\beta }}} \right)}_{0}}} \right]{{{\dot{q}}}_{\alpha }}{{{\dot{q}}}_{\beta }}}+\cdots \] \[\left( \begin{matrix} {{u}_{0}} \\ {{u}_{1}} \\ \vdots \\ {{u}_{N-1}} \\ \end{matrix} \right)=\sum\limits_{k>0}{\left[ \left( \begin{matrix} 1 \\ \cos ka \\ \vdots \\ \cos k\left( N-1 \right)a \\ \end{matrix} \right)\underbrace{{{C}_{k+}}\cos \left( {{\omega }_{k}}t+{{\varphi }_{k+}} \right)}_{\frac{2}{\sqrt{N}}{{q}_{k+}}}+\left( \begin{matrix} 0 \\ \sin ka \\ \vdots \\ \sin k\left( N-1 \right)a \\ \end{matrix} \right)\underbrace{{{C}_{k-}}\cos \left( {{\omega }_{k}}t+{{\varphi }_{k-}} \right)}_{\frac{2}{\sqrt{N}}{{q}_{k-}}} \right]}\] \[G(\vec{r},{\vec{r}}',\tau )=\int _{-\infty }^{\infty }\tilde{G}(\vec{r},{\vec{r}}',\omega )e^{-i \tau \omega }d\omega=\int_{-\infty }^{\infty } \frac{e^{-i \tau \omega } e^{i k |\vec{r}-{\vec{r}}'| }}{(2 \pi ) |\vec{r}-{\vec{r}}'| } \, d\omega=\frac{\delta \left(\tau -\frac{R}{c}\right)}{|\vec{r}-{\vec{r}}'| }\] \[ \begin{split} \mathcal{F}^{-1}(\ket{j}) &{}=\frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1}\exp\left(-2\uppi \ii \frac{jk}{2^n}\right)\ket{k}.\\ &{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\exp\left(-2\uppi \ii j\sum_{l=0}^{n-1}\frac{2^l k_l}{2^n}\right)\ket{k_{n-1}\cdots k_0}\\ &{}=\frac{1}{\sqrt{2^n}}\sum_{k_{n-1}=0}^1\cdots\sum_{k_{0}=0}^1\bigotimes_{l=1}^n\left[\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\ket{k_{n-l}}\right]\\ &{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\sum_{k_{n-l}=0}^1\exp\left(-2\uppi \ii j\frac{k_{n-l}}{2^l}\right)\ket{k_{n-l}}\right]\\ &{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\ket{0}_{n-l}+\ee^{-2\uppi \ii j /2^l}\ket{1}_{n-l}\right]\\ &{}=\frac{1}{\sqrt{2^n}}\bigotimes_{l=1}^n\left[\ket{0}_{n-l}+\ee^{-2\uppi \ii ({0.j_{l-1}\ldots j_0})}\ket{1}_{n-l}\right]. \end{split} \] \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 \[\begin{split}U\lb x_f,t_f;x_i,t_i\rb=&\sqrt{\frac{m\omega}{2\piup \ii \hbar\sin\left[\omega\lb t_f-t_i\rb\right]}}\\&{}\times\exp\left\{\frac{\ii m\omega}{2\hbar\sin\left[\omega \lb t_f-t_i\rb\right]}\left[\lb {x_i}^2+{x_f}^2\rb\cos\left[\omega\lb t_f-t_i\rb\right]-2 x_i x_f\right]\right\}\ed\end{split}\] \fi \end{document}