Add a latex sample to demo

Author: runarberg

demo/eulers-identity.latex.md

@@ -0,0 +1,97 @@
+# Euler’s identity
+
+In mathematics, **Euler’s identity** (also known as _Euler’s
+equation_) is the equality
+
+$$ e^{i\pi} + 1 = 0 $$
+
+where
+
+- $e$ is Euler's number, the base of natural logarithms,
+- $i$ is the imaginary unit, which by definition satisfies $i^2 = -1$,
+  and
+- $\pi$ is pi, the ratio of the circumference of a circle to its
+  diameter.
+
+Euler’s identity is named after the Swiss mathematician Leonhard
+Euler. It is a special case of Euler’s formula $e^{ix} = \cos x +
+i\sin x$ when evaluated for $x=\pi$. Euler's identity is considered an
+exemplar of mathematical beauty, as it shows a profound connection
+between the most fundamental numbers in mathematics. In addition, it
+is directly used in a proof that $\pi$ is transcendental, which
+implies the impossibility of squaring the circle.
+
+## Imaginary exponents
+
+Euler’s identity asserts that $e^{i\pi}$ is equal to $-1$. The
+expression $e^{i}$ is a special case of the expression $e^z$, where
+$z$ is any complex number. In general, $e^z$ is defined for complex
+$z$ by extending one of the definitions of the exponential function
+from real exponents to complex exponents. For example, one common
+definition is:
+
+$$
+e^z = \lim_{n\to\infty} \left(1 + \frac{z}{n}\right)^n.
+$$
+
+Euler’s identity therefore states that the limit, as $n$ approaches
+infinity, of $\left(1 + \frac{i \pi}{n}\right)^n$ is equal to $-1$.
+
+Euler’s identity is a special case of Euler’s formula, which states
+that for any real number $x$,
+
+$$ e^{ix} = \cos x + i\sin x $$
+
+where the inputs of the trigonometric functions sine and cosine are
+given in radians.
+
+![Euler's formula for a general angle](https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/330px-Euler%27s_formula.svg.png)
+
+In particular, when $x = \pi$,
+
+$$
+e^{i\pi} = \cos \pi + i \sin \pi.
+$$
+
+Since
+$$ \cos \pi = -1 $$
+and
+$$ \sin \pi = 0 $$
+
+it follows that
+$$ e^{i\pi} = -1 + 0i, $$
+
+which yields Euler's identity:
+
+$$ e^{i\pi} + 1 = 0. $$
+
+## Generalizations
+
+Euler’s identity is also a special case of the more general identity
+that the $n$th roots of unity, for $n > 1$, add up to $0$:
+
+$$
+\sum_{k=0}^{n-1} e^{2\pi i\frac{k}{n}} = 0 .
+$$
+
+Euler’s identity is the case where $n = 2$.
+
+A similar identity also applies to quaternion exponential: let $\left\{i, j,
+k\right\}$ be the basis quaternions; then,
+
+$$
+e^{\frac{1}{\sqrt{3}} (i \pm j \pm k) \pi} + 1 = 0 .
+$$
+
+More generally, let q be a quaternion with a zero real part and a norm
+equal to $1$; that is, $q = ai + bj + ck$, with $a^2 + b^2 + c^2 =
+1$. Then one has
+
+$$
+e^{q \pi} + 1 = 0 .
+$$
+
+The same formula applies to octonions, with a zero real part and a
+norm equal to $1$. These formulas are a direct generalization of
+Euler’s identity, since $i$ and $-i$ are the only complex numbers with
+a zero real part and a norm (absolute value) equal to $1$.

demo/main.js

@@ -37,6 +37,14 @@ if (
   throw new Error("playground not in DOM");
 }
 
+{
+  const { searchParams } = new URL(window.location.href);
+
+  if (searchParams.has("latex")) {
+    rendererControl.value = "temml";
+  }
+}
+
 let renderer = rendererControl.value === "temml" ? "temml" : "mathml";
 
 /**
@@ -222,7 +230,12 @@ blockDelimitersInput.addEventListener("change", handleBlockDelimitersChange);
 handleBlockDelimitersChange();
 
 if (!textInput.value) {
-  fetch("./eulers-identity.md")
+  const startDocPath =
+    rendererControl.value === "temml"
+      ? "./eulers-identity.latex.md"
+      : "./eulers-identity.md";
+
+  fetch(startDocPath)
     .then((r) => r.text())
     .then((text) => {
       textInput.value = text;