Hey everyone,
At the [last meetup] on [2025-06-11 Wed] we discussed Karthik &
Timothy’s new `LaTeX' preview feature ([repository]). One enhancement
is an adjusted baseline of formulas rendered and embedded as `SVG' in
`HTML' export, which makes it a viable `JavaScript'-free option. I
asked about exporting `MathML' as another option for `HTML', and Ihor
directed me towards the `ODT' export as a starting point.
Before going into details, why might you prefer `MathML' for equations
in `HTML'?
⁃ It “degrades” gracefully compared to `LaTeX' markup:
⁃ in the absence of `MathJax', major browsers [support] `MathML'
(`MathML Core' at least),
⁃ `MathJax' [supports] `MathML' and provides [accessibility]
[extensions] (this made me rethink my stance on `JS').
⁃ It integrates closely with `HTML':
⁃ baseline alignment is handled by the browser,
⁃ “dark mode” works automatically,
⁃ the math font can be matched to the text through `CSS',
⁃ math elements can also be styled with `CSS', e.g. colour all
square roots:
#+begin_src css
msqrt { color: red; }
#+end_src
⁃ That’s what `MathML' was built for ;)
The following is a summary of a few approaches I took and the
challenges I faced. Attached is an `org' file with equations in `TeX'
format gathered from various `MathML' “torture” tests if you want to
try it yourself. Please let me know what you think!
[last meetup] <https://list.orgmode.org/878ql1bt4s.fsf@localhost/>
[repository] <https://code.tecosaur.net/tec/org-mode/>
[support] <https://caniuse.com/mathml>
[supports] <https://docs.mathjax.org/en/latest/input/mathml.html>
[accessibility]
<https://docs.mathjax.org/en/latest/basic/a11y-extensions.html>
[extensions]
<https://docs.mathjax.org/en/latest/options/accessibility.html>
Conversion
══════════
There are several libraries that convert from `TeX' to `MathML', but I
will only mention a few:
1. [LaTeXML]
⁃ behind the [DLMF], [ar5iv], and the [experimental HTML] format on
the `arχiv',
2. [tex4ht],
⁃ part of `TeX' distributions,
3. [pandoc].
To produce the best output, these generally want to act on a complete
`TeX' file. However, you might want to have `org' handle the `HTML'
export and limit the converters to math fragments instead:
1. `latexmlmath' is exactly for this, and `latexmlc' can be convinced
to do the same:
#+begin_src shell
latexmlc 'literal:\[\sqrt{1+x^2}\]' --profile=math --presentationmathml
#+end_src
2. `make4ht' can read from standard input if you wrap it in a minimal
document, but as far as I can tell it can’t write fragments to
standard output:
#+begin_src shell
make4ht - "mathml" <<< "\documentclass{standalone}
\begin{document}\[\sqrt{1+x^2}\]\end{document}"
#+end_src
3. `pandoc' wraps the output in a paragraph, but otherwise works:
#+begin_src shell
pandoc -f latex -t html5 --mathml <<< '\[\sqrt{1+x^2}\]' | sed 's/<\/\?p>//g'
#+end_src
[LaTeXML] <https://math.nist.gov/~BMiller/LaTeXML/>
[DLMF] <https://dlmf.nist.gov/>
[ar5iv] <https://ar5iv.labs.arxiv.org/>
[experimental HTML] <https://info.arxiv.org/about/accessible_HTML.html>
[tex4ht] <https://tug.org/tex4ht/>
[pandoc] <https://pandoc.org/>
Math Fragments
══════════════
When `org-html-with-latex' is `html', the
`org-latex-to-html-convert-command' is invoked on `LaTeX' fragments to
produce `MathML' which is embedded in the `HTML', e.g.
#+begin_src emacs-lisp
(setopt org-html-with-latex 'html
org-latex-to-html-convert-command
"latexmlc literal:%i --profile=math --preload=amscd.sty -pmml
2>/dev/null")
#+end_src
This works really well across all the browsers I’ve tested. Some of
the drawbacks include:
⁃ With `org-html-with-latex' equal to `html' you’re forced to give up
the `MathJax' script. Since `MathJax' cooperates with `MathML', you
might still want the script to be embedded (or maybe conditionally
loaded, as in [LaTeXML-maybeMathjax.js]).
⁃ `AMS' environments like `equation' are not touched by
`org-latex-to-html-convert-command', in contrast to the
`org-latex-to-mathml-convert-command' used in `ODT' export:
#+begin_src emacs-lisp
(setopt org-latex-to-mathml-convert-command
"latexmlmath %i --presentationmathml=%o")
#+end_src
This leaves the raw `AMS' environments in the `HTML', and by default
there’s no `MathJax' to smooth things over.
⁃ Conversion has no knowledge of the preamble, and has to be part of
the command in a `--preload' or a `--preamble'.
⁃ Processing is slow, i.e. a simple formula takes on the order of a
few seconds for me. You can speed up subsequent builds by using the
[latexmls plugin], or something like [latexml-runner], but these are
unofficial scripts that aren’t distributed widely. The `MathML'
fragments should be cached on the `emacs' side to guard against this
(see the next approach).
⁃ Since we’re acting on fragments, content markup contains duplicate
`IDs' which trips up accessibility validation.
[LaTeXML-maybeMathjax.js]
<https://github.com/brucemiller/LaTeXML/blob/master/lib/LaTeXML/resources/javascript/LaTeXML-maybeMathjax.js>
[latexmls plugin] <https://github.com/dginev/LaTeXML-Plugin-latexmls/>
[latexml-runner] <https://github.com/dginev/latexml-runner>
Preview backend
═══════════════
Another way to approach this is to add a `mathml' entry to
`org-preview-latex-process-alist':
#+begin_src emacs-lisp
(setopt org-preview-latex-process-alist
'((mathml :program ("latexml" "xmllint")
:description "LaTeX → MathML"
:message "You need to install LaTeXML and XmlLint."
:image-input-type "html"
:image-output-type "mathml"
:latex-compiler
("latexmlc --presentationmathml --log=%f.log --base=%o
--destination=%O %f")
:image-converter
("xmllint --html --xpath '//math' %f > %O")
:post-clean (".tex" ".html" ".css" ".log"))))
#+end_src
Then, with the following value of `org-html-with-latex', `.mathml'
files are generated in the `ltximg' folder:
#+begin_src emacs-lisp
(setopt org-html-with-latex 'mathml)
#+end_src
This /almost/ works, but images created this way will be wrapped in an
`<img>' tag when we need to insert the `MathML' file verbatim. As far
as I can tell there’s no official way to change this behaviour, but as
a proof of concept we can advise `org-html--format-image':
#+begin_src emacs-lisp
(defun org-html--embed-image (source _ _)
"Return contents of the SOURCE with excess whitespace trimmed."
(string-trim (f-read-text source)))
(advice-add 'org-html--format-image :override 'org-html--embed-image)
#+end_src
Hacks aside, this works even better than the math fragment approach.
⁃ We still don’t have `MathJax'.
⁃ `AMS' environments are converted correctly.
⁃ The preamble is accounted for.
⁃ `MathML' output is cached, so subsequent exports are very fast.
⁃ We’re still acting on fragments, so content markup contains
duplicate `IDs'.
On the other hand, since the math colour is explicitly set when
generated this way, “dark mode” / inversion is broken.
Use `LaTeX' as an intermediary
══════════════════════════════
Finally, we could use these tools as intended and export `HTML' by
acting on an intermediate `LaTeX' format. This can be done with a
simple derived backend, where the conversion happens in
`org-html-mathml-compile' using the variable
`org-html-mathml-convert-command':
#+begin_src emacs-lisp
(org-export-define-derived-backend 'html-mathml 'latex
:menu-entry '(?h "Export to HTML+MathML"
((?m "As HTML+MathML file" org-html-mathml-export-to-file)
(?n "As HTML+MathML file and open"
(lambda (a s v b)
(if a (org-html-mathml-export-to-file t s v b)
(org-open-file (org-html-mathml-export-to-file nil
s v b))))))))
(defun org-html-mathml-export-to-file
(&optional async subtreep visible-only body-only ext-plist)
"Export current buffer to LaTeX then process through to HTML+MathML.
Return the HTML file’s name."
(interactive)
(let ((texfile (org-export-output-file-name ".tex" subtreep)))
(org-export-to-file 'latex
texfile async subtreep visible-only body-only ext-plist
#'org-html-mathml-compile)))
(defun org-html-mathml-compile (texfile)
"Compile a TeX file to HTML+MathML.
TEXFILE is the name of the file being compiled. Processing is done
through the command specified in `org-html-mathml-convert-command'.
Output is redirected to a \"*Org HTML+MathML Output*\" buffer.
Return HTML file name or raise an error if it couldn’t be produced."
(let* ((convert-template org-html-mathml-convert-command)
(log-buffer-name "*Org HTML+MathML Output*")
(log-buffer (get-buffer-create log-buffer-name))
(outfile (with-current-buffer log-buffer
(erase-buffer)
(message "Processing LaTeX file %s..." texfile)
(org-compile-file texfile convert-template "html"
(format "See %S for details"
log-buffer-name)
log-buffer))))
outfile))
#+end_src
Here are some settings for the three conversion tools I mentioned:
#+begin_src emacs-lisp
(setopt org-html-mathml-convert-command
'("latexmlc --presentationmathml --destination=%O %f"))
(setopt org-html-mathml-convert-command
'("make4ht %f \"mathml\""))
(setopt org-html-mathml-convert-command
'("pandoc -f latex -t html5 --mathml %f > %O"))
#+end_src
In my testing `LaTeXML' output looked the best, followed closely by
`tex4ht' (the `pandoc' output had several artifacts).
Best,
--
Jacob S. Gordon
jacob.as.gor...@gmail.com
Please avoid sending me HTML emails and MS Office documents.
https://useplaintext.email/#etiquette#+TITLE: ~MathML~ test document
#+HTML_DOCTYPE: html5
#+LATEX_HEADER: \usepackage{amscd}
* ~W3C~ ~MathML~ test suite
The following equations are from the ~TortureTests.Complexity~ section of the [[https://www.w3.org/Math/testsuite/][W3C MathML test suite]].
+ Bernoulli trials: \(P(E) = \binom{n}{k}p^k(1-p)^{n-k}\).
+ Cauchy–Schwarz inequality:
\[ \left( \sum_{k = 1}^na_kb_k \right)^2 \le \left( \sum_{k = 1}^na_k \right)^2\left( \sum_{k = 1}^nb_k \right)^2. \]
+ Cauchy integral formula:
\[ f(z)\cdot\text{Ind}_{\gamma}(z) = \frac{1}{2\pi i}\oint\limits_{\gamma} \frac{f(\xi)}{\xi-z}\mathrm{d}\xi. \]
+ Cross product:
\[ \mathbf{V}_1\times\mathbf{V}_2 =
\begin{vmatrix}
\hat{\imath} & \hat{\jmath} & \hat{k} \\
V_{1x} & V_{1y} & V_{1z} \\
V_{2x} & V_{2y} & V_{2z}\end{vmatrix}. \]o
+ Vandermonde determinant:
\[ \begin{vmatrix}
1 & 1 & \cdots & 1 \\
v_1 & v_2 & \cdots & v_n \\
v_1^2 & v_2^2 & \cdots & v_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
v_1^{n-1} & v_2^{n-1} & \cdots & v_n^{n-1}
\end{vmatrix} = \prod_{\substack{1 \le i \le n \\ 1 \le j \le n}}(v_i - v_j). \]
+ Lorenz equations:
\[ \begin{split}
\dot{x} &= \sigma(y - x), \\
\dot{y} &= x(\rho - z) - y, \\
\dot{z} &= x y -\beta z.
\end{split} \]
+ Maxwell’s equations:
\[\left\{ \begin{aligned}
\nabla\cdot\vec{\mathbf{D}} &= \rho_f, \\
\nabla\times\vec{\mathbf{E}} + \frac{\partial\vec{\mathbf{B}}}{\partial t} &= \vec{\mathbf{0}}, \\
\nabla\cdot\vec{\mathbf{B}} &= 0, \\
\nabla\times\vec{\mathbf{H}} - \frac{\partial\vec{\mathbf{D}}}{\partial t} &= \vec{\mathbf{J}}_f.
\end{aligned}\right. \]
+ Einstein’s field equations: \(G_{\mu\nu} + \Lambda g_{\mu\nu} = R_{\mu\nu} + \left(\Lambda - \frac{1}{2}R\right)g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}.\)
+ Rogers–Ramanujan continued fraction:
\[ \frac{1}{(\sqrt{\varphi\sqrt{5}}-\varphi)e^{\frac{2\pi}{5}}} = 1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-8\pi}}{1 + \ddots}}} = \frac{e^{-\frac{2\pi}{5}}}{R(e^{-2\pi})}. \]
+ Another Ramanujan identity:
\[ \sum_{k = 1}^{\infty} = \frac{1}{2^{\lfloor k\cdot\varphi \rfloor}} = \cfrac{1}{2^0 + \cfrac{1}{2^1 + \ddots}}. \]
+ Rogers–Ramanujan identity:
\[ 1 + \sum_{k = 1}^{\infty} \frac{q^{k^2+k}}{(1-q)(1-q^2)\cdots(1-q^k)} = \prod_{j = 0}^{\infty}\frac{1}{(1-q^{5j+2})(1 - q^{5j+3})},\;\text{for}\;|q| < 1. \]
+ Commutative diagram:
\[ \begin{array}{ccc}
H & \longleftarrow & K \\
\downarrow & & \uparrow \\
H & \longrightarrow & K
\end{array}, \]
\[ \begin{CD}
H @<<< K \\
@VVV @AAA \\
H @>>> K \end{CD}. \]
* Mozilla ~MathML~ test
The following equations are from the [[https://www-archive.mozilla.org/projects/mathml/demo/texvsmml.xhtml][Mozilla MathML “torture” test suite]], which are mostly from the indicated sections of the ~TeXbook~.
[[https://fred-wang.github.io/MathFonts/mozilla_mathml_test/][Fred Wang]] and [[https://pshihn.github.io/math-ml/examples/torture.html][Preet Shihn]] provide similar pages with the same tests.
+ Chapter 16:
\[ x^2y^2, \]
\[ {}_2F_3. \]
+ Chapter 17:
\[ \frac{x + y^2}{k+1}, \]
\[ x + y^{\frac{2}{k+1}}, \]
\[ \frac{a}{b/2}, \]
\[ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4}}}}, \]
\[ a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \frac{1}{a_4}}}}, \]
\[ \binom{n}{k/2}. \]
+ Exercise 17.7:
\[ \binom{p}{2}x^2y^{p-2} - \frac{1}{\mathstrut 1-x}\frac{1}{\mathstrut 1-x^2}. \]
+ Chapter 17:
\[ \sum_{\substack{0 \le i \le m \\ 0 < j < n}}P(i,j). \]
+ Chapter 16:
\[ x^{2y}. \]
+ Exercise 17.8:
\[ \sum_{i = 1}^p\sum_{j = 1}^q\sum_{k = 1}^ra_{ij}b_{jk}c_{ki}. \]
+ Chapter 17:
\[ \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}}}. \]
+ Exercise 17.10:
\[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \bigl|\varphi(x+iy)\bigr|^2 = 0. \]
+ Chapter 16:
\[ 2^{2^{2^x}}. \]
+ Chapter 18:
\[ \int_1^x \frac{\mathrm{d}t}{t}, \]
\[ \int\!\!\!\int_D \mathrm{d}x\,\mathrm{d}y. \]
+ Exercise 18.23:
\[ f(x) = \begin{cases}
1/3 &\text{if } 0 \le x \le 1, \\
2/3 &\text{if } 3 \le x \le 4, \\
0 &\text{elsewhere},
\end{cases}. \]
+ Chapter 18:
\[ \overbrace{x+\cdots+x}^{k\;\text{times}}. \]
+ Exercise 18.40:
\[ \sum_{p\;\text{prime}}f(p) = \int_{t > 1} f(t)\,\mathrm{d}\pi(t). \]
+ Exercise 18.41:
\[ \{ \underbrace{\overbrace{\mathstrut a,\ldots,a}^{k\;a\text{'s}},\overbrace{\mathstrut b,\ldots,b}^{l\; b\text{'s}}}_{k+l\;\text{elements}}\}. \]
+ Exercise 18.42:
\[ \begin{pmatrix}
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix} &
\begin{pmatrix}
e & f \\ g & h
\end{pmatrix} \\
0 &
\begin{pmatrix}
i & j \\ k & l
\end{pmatrix} \\
\end{pmatrix}. \]
+ Exercise 18.43:
\[ \det\left|\,\begin{matrix}
c_0 & c_1\hfill & c_2\hfill & \ldots & c_n\hfill \\
c_1 & c_2\hfill & c_3\hfill & \ldots & c_{n+1}\hfill \\
c_2 & c_3\hfill & c_4\hfill & \ldots & c_{n+2}\hfill \\
\,\vdots\hfill & \,\vdots\hfill & \,\vdots\hfill && \,\vdots\hfill \\
c_n & c_{n+1}\hfill & c_{n+2}\hfill & \ldots & c_{2n}\hfill
\end{matrix}\right| > 0. \]
+ Chapter 16:
\[ y_{x^2}, \]
\[ x_{92}^{31415} + \pi, \]
\[ x_{y_b^a}^{z_c^d}, \]
\[ y_3^{\prime\prime\prime}. \]
+ Stirling’s approximation:
\[ \lim_{n \rightarrow \infty}\frac{\sqrt{2\pi n}}{n!}\left(\frac{n}{e}\right)^n = 1. \]
+ Leibniz formula for the determinant:
\[ \det(A) = \sum_{\sigma \in S_n} \epsilon(\sigma) \prod_{i = 1}^na_{i,\sigma_i}. \]
* ~MathML~ browser test
The following equations are from the [[http://eyeasme.com/Joe/MathML/MathML_browser_test.html][MathML browser test (presentation markup)]].
+ [[http://en.wikipedia.org/wiki/Axiom_of_power_set][Axiom of power set]]:
\[ \forall A\; \exists P\; \forall B\; [ B \in P \iff \forall C (C \in B \implies C \in A)]. \]
+ [[http://en.wikipedia.org/wiki/Demorgans_law][De Morgan’s law]]:
+ Logic: \(\neg(p \land q) \iff (\neg p) \lor (\neg q),\)
+ Boolean algebra: \(\overline{\cup_{i = 1}^nA_i} = \cap_{i = 1}^n\overline{A_i}.\)
+ [[http://en.wikipedia.org/wiki/Quadratic_equation][Quadratic formula]]:
\[ x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}. \]
+ [[http://en.wikipedia.org/wiki/Combination][Binomial coefficient]]:
\[ C(n,k) = C_n^k = {}_nC_k = \binom{n}{k} = \frac{n!}{k!(n-k)!}. \]
+ [[http://en.wikipedia.org/wiki/Sophomore's_dream][Sophmore’s dream]]:
\[ \int_0^1 x^x\,\mathrm{d}x = \sum_{n = 1}^{\infty}\frac{(-1)^{n+1}}{n^n}. \]
+ [[http://en.wikipedia.org/wiki/Divergence][Divergence]]: \(\nabla\cdot\vec{v} = \frac{\partial v_{\mathstrut x}}{\partial x} + \frac{\partial v_{\mathstrut y}}{\partial y} + \frac{\partial v_{\mathstrut z}}{\partial z}.\)
+ [[http://en.wikipedia.org/wiki/Complex_number][Complex number]]:
\[ c = \overbrace{\underbrace{a}_{\text{real}} + \underbrace{i b}_{\text{imaginary}} }^{\text{complex number}}. \]
+ [[http://en.wikipedia.org/wiki/Moore_matrix][Moore matrix]]:
\[ M = \begin{bmatrix}
\alpha_1 & \alpha_1^q & \ldots & \alpha_1^{q^{n-1}} \\
\alpha_2 & \alpha_2^q & \ldots & \alpha_2^{q^{n-1}} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha_m & \alpha_m^q & \ldots & \alpha_m^{q^{n-1}} \\
\end{bmatrix}.
\]
+ [[http://en.wikipedia.org/wiki/Sphere][Sphere volume]]: The region occupied by a sphere of radius \(R\) in spherical coordinates is \(S = \{0 \le \varphi < 2\pi,\; 0 \le \theta \le \pi,\; 0 \le \rho \le R\},\) so we can calculate the volume to be \(\frac{4}{3}\pi R^3:\)
\[ \begin{split}
\text{Volume} &= \iiint_S \rho^2\sin\theta\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\varphi, \\
&= \int_0^{2\pi}\mathrm{d}\varphi \int_0^{\pi}\sin\theta\,\mathrm{d}\theta\int_0^R\rho^2\,\mathrm{d}\rho, \\
&= 2\pi [\cos\theta]_{\pi}^0\left[\frac{1}{3}\rho^3\right]_0^R, \\
&= \frac{4}{3}\pi R^3.
\end{split} \]
+ [[http://en.wikipedia.org/wiki/Schwinger-Dyson_equation][Schwinger–Dyson equation]]:
\[ \left\langle \psi \left\vert \mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi] \right\} \right\vert \psi \right\rangle = -i \left\langle \psi \left\vert \mathcal{T}\left\{F[\phi]\frac{\delta}{\delta\phi}S[\phi] \right\} \right| \psi \right\rangle. \]
+ [[http://en.wikipedia.org/wiki/Differentiable_manifold][Tangent vector of differentiable manifold]]:
\[ \gamma_1 \equiv \gamma_2 \quad\iff\quad \begin{cases}
\gamma_1(0) = \gamma_2(0) = p,\text{ and} \\
\frac{\mathrm{d}}{\mathrm{d}t}\phi\circ\gamma_1(t)\rvert_{t = 0} = \frac{\mathrm{d}}{\mathrm{d}t}\phi\circ\gamma_2(t)\rvert_{t = 0}
\end{cases}.
\]
+ [[http://en.wikipedia.org/wiki/Cichon's_diagram][Cichoń’s diagram]]:
\[\begin{array}{ccccccccccc}
& & \text{cov}(\mathcal{L}) & \rightarrow & \mathrm{non}(\mathcal{K}) & \rightarrow & \mathrm{cof}(\mathcal{K}) & \rightarrow & \mathrm{cof}(\mathcal{L}) & \rightarrow & 2^{\aleph_0} \\
& & & & \uparrow & & \uparrow & & & & \\
& & \Bigg\uparrow & & \mathfrak{b} & \rightarrow & \mathfrak{d} & & \Bigg\uparrow & & \\
& & & & \uparrow & & \uparrow & & & & \\
\aleph_1 & \rightarrow & \mathrm{add}(\mathcal{L}) & \rightarrow & \mathrm{add}(\mathcal{K}) & \rightarrow & \mathrm{cov}(\mathcal{K}) & \rightarrow & \mathrm{non}(\mathcal{L}) & &
\end{array}.\]
+ Multiscripts & Greek alphabet:
\[ \sideset{_{{}_{\alpha}^{\beta}\mathfrak{A}_{\delta}^{\gamma}}^{{}_{\epsilon}^{\zeta}\mathfrak{B}_{\theta}^{\eta}}}{_{{}_{\rho}^{\sigma}\mathfrak{E}_{\upsilon}^{\tau}}^{{_{\nu}^{\xi}\mathfrak{D}_{\pi}^o}}}\prod_{{}_{\phi}^{\chi}\mathfrak{F}_{\omega}^{\psi}}^{{}_{\iota}^{\kappa}\mathfrak{C}_{\mu}^{\lambda}}. \]
+ Nested roots:
\[ \frac{\sqrt{1 + \sqrt[3]{2 + \sqrt[5]{3 + \sqrt[7]{4 + \sqrt[11]{5 + \sqrt[13]{6 + \sqrt[17]{7 + \sqrt[19]{A}}}}}}}}}{e^{\pi}} = x^{\prime\prime\prime}. \]
+ Display sizes:
\[ \displaystyle x^2 + y^2,\;\textstyle x^2 + y^2,\;\scriptstyle x^2 + y^2,\;\scriptscriptstyle x^2+y^2. \]
* Miscellaneous
The following equations are all taken from the ~TeXbook~.
+ Chapter 17:
\[ \frac{a+1}{b}\bigg/\frac{c+1}{d}. \]
+ Exercise 17.13:
\[ \pi(n) = \sum_{k = 2}^{n}\left\lfloor \frac{\phi(k)}{k-1}\right\rfloor. \]
+ Exercise 17.14:
\[ \pi(n) = \sum_{m = 2}^n \left\lfloor \left( \sum_{k = 1}^{m-1}\lfloor(m/k)\big/\lceil m/k\rceil\rfloor\right)^{-1} \right\rfloor. \]
+ Exercise 18.2:
\[ p_1(n) = \lim_{m\rightarrow\infty}\sum_{\nu = 0}^{\infty}\left(1 - \cos^{2m}(\nu!^n\pi/n) \right). \]
+ Exercise 18.5:
\[ \binom{n}{k} \equiv \binom{\lfloor n/p\rfloor}{\lfloor k/p\rfloor}\binom{n\bmod p}{k\bmod p}\pmod p. \]
+ Exercise 18.10: Let \(H\) be a Hilbert space, \(C\) a closed bounded convex subset of \(H,\) \(T\) a nonexpansive self map of \(C.\)
Suppose that as \(n\rightarrow\infty,\) \(a_{n,k} \rightarrow 0\) for each \(k,\) and \(\gamma_n = \sum_{k = 0}^{\infty}(a_{n,k+1}-a_{n,k})^+ \rightarrow 0.\)
Then for each \(x\) in \(C,\) \(A_nx = \sum_{k = 0}^{\infty}a_{n,k}T^kx\) converges weakly to a fixed point of \(T.\)
+ Exercise 18.11:
\[ \int_0^{\infty} \frac{t - i b}{t^2+b^2}e^{i a t}\;\mathrm{d}t = e^{a b}E_1(a b),\quad a,b > 0. \]
+ Exercise 18.37:
\[ \prod_{j \ge 0} \left(\sum_{k \ge 0}a_{j k}z^k\right) = \sum_{n \ge 0}z^n\left(\sum_{\substack{k_0,k_1,\ldots \ge 0 \\ k_0 + k_1 + \cdots = n}} a_{0k_0}a_{1k_1}\ldots\right). \]
+ Exercise 18.38:
\[ \frac{(n_1+n_2+\cdots+n_m)!}{n_1!n_2!\ldots n_m!} = { n_1 + n_2 \choose n_2 } { n_1 + n_2 + n_3 \choose n_3 } \cdots { n_1 + n_2 + \cdots + n_m \choose n_m }. \]
+ Exercise 19.5:
\[ \prod_{k \ge 0}\frac{1}{1-q^k z} = \sum_{n \ge 0}z^n\Bigg / \prod_{1 \le k \le n}(1 - q^k). \]
+ Exercise 19.13:
\[\begin{split} \left(\int_{-\infty}^{+\infty}e^{-x^2}\;\mathrm{d}x\right)^2 &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-(x^2+y^2)}\;\mathrm{d}x\,\mathrm{d}y, \\
&= \int_0^{2\pi}\int_0^{\infty}e^{-r^2}r\;\mathrm{d}r\,\mathrm{d}\theta, \\
&= 2\pi \left[\frac{1}{2}e^{-r^2}\right]_{\infty}^0, \\
&= \pi. \end{split} \]
* ~AMS~ math environments
These examples are from /The =LaTeX= Companion, 3ed/, Chapter 11.
+ ~equation*~
\begin{equation*}
n^2 + m^2 = k^2
\end{equation*}
+ ~equation~:
\begin{equation}
x^2 = y^2 = z^2
\end{equation}
+ ~equation+split~:
\begin{equation}
\begin{split}
(a + b)^4 &= (a + b)^2(a + b)^2 \\
&= (a^2 + 2 a b + b^2)(a^2 + 2 a b + b^2) \\
&= a^4 + 4a^3b + 6a^2b^2 + 4a b^2 + b^4
\end{split}
\end{equation}
+ ~align~:
\begin{align}
x^2 + y^2 &= z^2 \\ x^3 + y^3 &< z^3
\end{align}
\begin{align}
x^2 + y^2 &= 1 & x^3 + y^3 &= 1 \\
x &= \sqrt{1-y^2} & x &= \sqrt[3]{1-y^3}
\end{align}
\begin{align}
x &= y & X &= Y & a &= b + c \\
x + x^{\prime} &= y + y^{\prime} & X + X^{\prime} &= Y + Y^{\prime} & a^{\prime}b &= c^{\prime}b
\end{align}
\begin{align}
A_1 &= N_0(\lambda ; \Omega^{\prime}) - \phi(\lambda ; \Omega^{\prime}) \\
A_2 &= \phi(\lambda ; \Omega^{\prime})\phi(\lambda ; \Omega) \\
\intertext{and finally}
A_3 &= \mathcal{N}(\lambda ; \omega)
\end{align}
+ ~align*~:
\begin{align*}
(a + b)^3 &= (a + b)(a + b)^2 \\
&= (a + b)(a^2 + 2 a b + b^2) \\
&= a^3 + 3a^2b + 3a b^2 + b^3
\end{align*}
+ ~eqnarray~:
\begin{eqnarray}
x^2 + y^2 &= z^2 \\ x^3 + y^3 &< z^3
\end{eqnarray}
+ ~gather~:
\begin{gather}
(a + b)^2 = a^2 + 2 a b + b^2 \\
(a + b) \cdot (a - b) = a^2 - b^2
\end{gather}
+ ~subequations~:
\begin{subequations} \label{eq:1}
\begin{align}
f &= g \label{eq:1A} \\
f^{\prime} &= g^{\prime} \label{eq:1B} \\
\mathcal{L}f &= \mathcal{L}g \label{eq:1C}
\end{align}
\end{subequations}
+ ~boxed~:
\begin{equation}
\boxed{W_t - F \subseteq V(P_i) \subseteq W_t}
\end{equation}
* Raw ~MathML~
#+begin_export html
<math xmlns='http://www.w3.org/1998/Math/MathML' alttext='1+x+x^{2}+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x}' display='block'>
<mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mi mathvariant="normal">⋯</mi>
<mo>+</mo>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msup>
<mi>x</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</math>
#+end_export
