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diff --git a/doc/apple-by-example.html b/doc/apple-by-example.html
@@ -166,8 +166,10 @@
       margin: 0 0.8em 0.2em -1.6em;
       vertical-align: middle;
     }
-    .display.math{display: block; text-align: center; margin: 0.5rem auto;}
   </style>
+  <script
+  src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js"
+  type="text/javascript"></script>
 </head>
 <body>
 <header id="title-block-header">
@@ -194,6 +196,8 @@ <h1 class="title">Apple by Example</h1>
 <li><a href="#array-literals" id="toc-array-literals">Array
 Literals</a></li>
 <li><a href="#reverse" id="toc-reverse">Reverse</a></li>
+<li><a href="#outer-product" id="toc-outer-product">Outer
+Product</a></li>
 </ul></li>
 <li><a href="#examples" id="toc-examples">Examples</a>
 <ul>
@@ -213,6 +217,17 @@ <h1 class="title">Apple by Example</h1>
 <ul>
 <li><a href="#filter" id="toc-filter">Filter</a></li>
 </ul></li>
+<li><a href="#numerics" id="toc-numerics">Numerics</a>
+<ul>
+<li><a href="#arithmetic-geometric-mean"
+id="toc-arithmetic-geometric-mean">Arithmetic-Geometric Mean</a></li>
+<li><a href="#hypergeometric-function"
+id="toc-hypergeometric-function">Hypergeometric Function</a></li>
+</ul></li>
+<li><a href="#geography" id="toc-geography">Geography</a>
+<ul>
+<li><a href="#mercator" id="toc-mercator">Mercator</a></li>
+</ul></li>
 <li><a href="#number-theory" id="toc-number-theory">Number Theory</a>
 <ul>
 <li><a href="#primality-check" id="toc-primality-check">Primality
@@ -229,7 +244,7 @@ <h2 id="installation">Installation</h2>
 <p>The REPL and typechecker are available on the <a
 href="https://github.com/vmchale/apple/releases">release page</a>.</p>
 <h3 id="libraries">Libraries</h3>
-<p>To install the libraries, you will can use <a
+<p>To install the libraries, you can use <a
 href="https://www.haskell.org/ghcup/">ghcup</a> to install cabal and
 GHC.</p>
 <p>Then:</p>
@@ -303,6 +318,29 @@ <h2 id="reverse">Reverse</h2>
 [ [2, 4]
 , [1, 0]
 , [0, 1] ]</code></pre>
+<h2 id="outer-product">Outer Product</h2>
+<p>The outer product <code>⊗</code> creates a table by applying some
+function.</p>
+<p>It has type</p>
+<pre><code> &gt; :ty \f.\x.\y. x f⊗ y
+(a → b → c) → Arr sh0 a → Arr sh1 b → Arr (sh0 ⧺ sh1) c</code></pre>
+<pre><code> &gt; (frange 0 9 10) (*)⊗ (frange 0 9 10)
+[ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+, [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
+, [0.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0]
+, [0.0, 3.0, 6.0, 9.0, 12.0, 15.0, 18.0, 21.0, 24.0, 27.0]
+, [0.0, 4.0, 8.0, 12.0, 16.0, 20.0, 24.0, 28.0, 32.0, 36.0]
+, [0.0, 5.0, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0]
+, [0.0, 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0, 48.0, 54.0]
+, [0.0, 7.0, 14.0, 21.0, 28.0, 35.0, 42.0, 49.0, 56.0, 63.0]
+, [0.0, 8.0, 16.0, 24.0, 32.0, 40.0, 48.0, 56.0, 64.0, 72.0]
+, [0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0, 63.0, 72.0, 81.0] ]</code></pre>
+<pre><code> &gt; (frange 0 4 5) [(x,y)]⊗ (frange 0 4 5)
+[ [(0.0*0.0), (0.0*1.0), (0.0*2.0), (0.0*3.0), (0.0*4.0)]
+, [(1.0*0.0), (1.0*1.0), (1.0*2.0), (1.0*3.0), (1.0*4.0)]
+, [(2.0*0.0), (2.0*1.0), (2.0*2.0), (2.0*3.0), (2.0*4.0)]
+, [(3.0*0.0), (3.0*1.0), (3.0*2.0), (3.0*3.0), (3.0*4.0)]
+, [(4.0*0.0), (4.0*1.0), (4.0*2.0), (4.0*3.0), (4.0*4.0)] ]</code></pre>
 <h1 id="examples">Examples</h1>
 <h2 id="dot-product">Dot Product</h2>
 <pre><code>[(+)/ ((*)`((x::Vec n float)) y)]</code></pre>
@@ -333,8 +371,8 @@ <h2 id="rising-factorial">Rising Factorial</h2>
 <p>The <a
 href="https://mathworld.wolfram.com/RisingFactorial.html">rising
 factorial</a> is defined as:</p>
-<p><span
-class="math display"><em>x</em><sup>(<em>n</em>)</sup> = <em>x</em>(<em>x</em> + 1)⋯(<em>x</em> + <em>n</em> − 1)</span></p>
+<p><span class="math display">\[ x^{(n)} = x(x+1)\cdots
+(x+n-1)\]</span></p>
 <p>In apple this is</p>
 <pre><code>[(*)/ₒ 1 (⍳ x (x+y-1) 1)]</code></pre>
 <p><code>/ₒ</code> is a ternary operator, fold with seed.</p>
@@ -355,13 +393,54 @@ <h2 id="array">Array</h2>
 <pre><code>[}.\`7 x]</code></pre>
 <h3 id="filter">Filter</h3>
 <pre><code>\p.\xs. (xs˙)&#39;p⩪xs</code></pre>
+<h2 id="numerics">Numerics</h2>
+<h3 id="arithmetic-geometric-mean">Arithmetic-Geometric Mean</h3>
+<pre><code>λx.λy.([{a⟜x-&gt;1;g⟜x-&gt;2;((a+g)%2,√(a*g))}]^:6 (x,y))-&gt;1</code></pre>
+<p>Thence the <a
+href="https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind">complete
+elliptic integral of the first kind</a>:</p>
+<pre><code>λk.
+{
+  agm ← λx.λy.([{a⟜x-&gt;1;g⟜x-&gt;2;((a+g)%2,√(a*g))}]^:6 (x,y))-&gt;1;
+  𝜋%(2*agm 1 (√(1-k^2)))
+}</code></pre>
+<p>Gauss’ approximation of the logarithm:</p>
+<pre><code>λm.λx.
+{
+  amgm ← λx.λy.([{a⟜x-&gt;1;g⟜x-&gt;2;((a+g)%2,√(a*g))}]^:15 (x,y))-&gt;1;
+  -- m&gt;2
+  𝜋%(2*amgm 1 (0.5^(m-2)%x))-ℝ m*0.6931471805599453
+}</code></pre>
+<h3 id="hypergeometric-function">Hypergeometric Function</h3>
+<p><span class="math display">\[
+{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x) = \sum_{n=0}^\infty
+\frac{(a_1)_n\ldots (a_p)_n}{(b_1)_n\ldots
+(b_q)_n}\frac{x^n}{n!}\]</span></p>
+<p>where <span class="math inline">\((x)_n\)</span> is the <a
+href="#rising-factorial">rising factorial</a> above.</p>
+<pre><code>λa.λb.λz.
+{
+  rf ← [(*)/ₒ 1 (𝒻 x (x+y-1) (⌊y))]; fact ← rf 1;
+  Σ ← λN.λa. (+)/ₒ 0 (a&#39;(⍳ 0 N 1)); Π ← [(*)/x];
+  Σ 30 (λn. {nn⟜ℝ n; (Π ((λa.rf a nn)&#39;a)%Π((λb. rf b nn)&#39;b))*(z^n%fact nn)})
+}</code></pre>
+<p>We can use the REPL to inspect the type:</p>
+<pre><code> &gt; :yank H math/hypergeometric.🍏
+ &gt; :ty H
+Vec (i + 1) float → Vec (i + 1) float → float → float</code></pre>
+<h2 id="geography">Geography</h2>
+<h3 id="mercator">Mercator</h3>
+<p><span class="math display">\[ x \mapsto \lambda - \lambda_0 \]</span>
+<span class="math display">\[ y \mapsto \frac{1}{2}
+\log{\left(\frac{1+\sin \phi}{1-\sin \phi}\right)} \]</span></p>
+<pre><code>\𝜆₀.\𝜑.\𝜆.{a⟜sin.𝜑;(𝜆-𝜆₀,(_.((1+a)%(1-a)))%2)}</code></pre>
 <h2 id="number-theory">Number Theory</h2>
 <h3 id="primality-check">Primality Check</h3>
 <pre><code>λn.¬((∨)/ₒ #f ([(n|x)=0]&#39;(⍳ 2 (⌊(√(ℝn))) 1)))</code></pre>
 <h3 id="radical">Radical</h3>
 <p>Compute the radical of an integer <span
-class="math inline"><em>n</em></span>, <span
-class="math inline">∏<sub><em>p</em>|<em>n</em></sub><em>p</em></span></p>
+class="math inline">\(n\)</span>, <span
+class="math inline">\(\displaystyle \prod_{p|n} p\)</span></p>
 <pre><code>λn.
   { ni ⟜ ⌊(√(ℝn))
   ; isPrime ← λn.¬((∨)/ₒ #f ([(n|x)=0]&#39;(⍳ 2 (⌊(√(ℝn))) 1)))

diff --git a/doc/apple-by-example.md b/doc/apple-by-example.md
@@ -12,7 +12,7 @@ The REPL and typechecker are available on the [release page](https://github.com/
 
 ### Libraries
 
-To install the libraries, you will can use [ghcup](https://www.haskell.org/ghcup/) to install cabal and GHC.
+To install the libraries, you can use [ghcup](https://www.haskell.org/ghcup/) to install cabal and GHC.
 
 Then:
 
@@ -141,6 +141,40 @@ Reverse applied to a higher-dimensional array reverses elements (sub-arrays) alo
 , [0, 1] ]
 ```
 
+## Outer Product
+
+The outer product `⊗` creates a table by applying some function.
+
+It has type
+
+```
+ > :ty \f.\x.\y. x f⊗ y
+(a → b → c) → Arr sh0 a → Arr sh1 b → Arr (sh0 ⧺ sh1) c
+```
+
+```
+ > (frange 0 9 10) (*)⊗ (frange 0 9 10)
+[ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+, [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
+, [0.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0]
+, [0.0, 3.0, 6.0, 9.0, 12.0, 15.0, 18.0, 21.0, 24.0, 27.0]
+, [0.0, 4.0, 8.0, 12.0, 16.0, 20.0, 24.0, 28.0, 32.0, 36.0]
+, [0.0, 5.0, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0]
+, [0.0, 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0, 48.0, 54.0]
+, [0.0, 7.0, 14.0, 21.0, 28.0, 35.0, 42.0, 49.0, 56.0, 63.0]
+, [0.0, 8.0, 16.0, 24.0, 32.0, 40.0, 48.0, 56.0, 64.0, 72.0]
+, [0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0, 63.0, 72.0, 81.0] ]
+```
+
+```
+ > (frange 0 4 5) [(x,y)]⊗ (frange 0 4 5)
+[ [(0.0*0.0), (0.0*1.0), (0.0*2.0), (0.0*3.0), (0.0*4.0)]
+, [(1.0*0.0), (1.0*1.0), (1.0*2.0), (1.0*3.0), (1.0*4.0)]
+, [(2.0*0.0), (2.0*1.0), (2.0*2.0), (2.0*3.0), (2.0*4.0)]
+, [(3.0*0.0), (3.0*1.0), (3.0*2.0), (3.0*3.0), (3.0*4.0)]
+, [(4.0*0.0), (4.0*1.0), (4.0*2.0), (4.0*3.0), (4.0*4.0)] ]
+```
+
 # Examples
 
 ## Dot Product
@@ -233,6 +267,69 @@ To drop the first 6 elements:
 \p.\xs. (xs˙)'p⩪xs
 ```
 
+## Numerics
+
+### Arithmetic-Geometric Mean
+
+```
+λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1
+```
+
+Thence the [complete elliptic integral of the first kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind):
+
+```
+λk.
+{
+  agm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1;
+  𝜋%(2*agm 1 (√(1-k^2)))
+}
+```
+
+Gauss' approximation of the logarithm:
+
+```
+λm.λx.
+{
+  amgm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:15 (x,y))->1;
+  -- m>2
+  𝜋%(2*amgm 1 (0.5^(m-2)%x))-ℝ m*0.6931471805599453
+}
+```
+
+### Hypergeometric Function
+
+$$ {}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x) = \sum_{n=0}^\infty \frac{(a_1)_n\ldots (a_p)_n}{(b_1)_n\ldots (b_q)_n}\frac{x^n}{n!}$$
+
+where $(x)_n$ is the [rising factorial](#rising-factorial) above.
+
+```
+λa.λb.λz.
+{
+  rf ← [(*)/ₒ 1 (𝒻 x (x+y-1) (⌊y))]; fact ← rf 1;
+  Σ ← λN.λa. (+)/ₒ 0 (a'(⍳ 0 N 1)); Π ← [(*)/x];
+  Σ 30 (λn. {nn⟜ℝ n; (Π ((λa.rf a nn)'a)%Π((λb. rf b nn)'b))*(z^n%fact nn)})
+}
+```
+
+We can use the REPL to inspect the type:
+
+```
+ > :yank H math/hypergeometric.🍏
+ > :ty H
+Vec (i + 1) float → Vec (i + 1) float → float → float
+```
+
+## Geography
+
+### Mercator
+
+$$ x \mapsto \lambda - \lambda_0 $$
+$$ y \mapsto \frac{1}{2} \log{\left(\frac{1+\sin \phi}{1-\sin \phi}\right)} $$
+
+```
+\𝜆₀.\𝜑.\𝜆.{a⟜sin.𝜑;(𝜆-𝜆₀,(_.((1+a)%(1-a)))%2)}
+```
+
 ## Number Theory
 
 ### Primality Check
@@ -243,7 +340,7 @@ To drop the first 6 elements:
 
 ### Radical
 
-Compute the radical of an integer $n$, $\prod_{p|n} p$
+Compute the radical of an integer $n$, $\displaystyle \prod_{p|n} p$
 
 ```
 λn.

diff --git a/docs/index.html b/docs/index.html
@@ -0,0 +1 @@
+../doc/apple-by-example.html