From f2a71e3581019332c56f3568a47f1f8b47d826e5 Mon Sep 17 00:00:00 2001
From: Karan Anand <anandkarancompsci@gmail.com>
Date: Thu, 27 Aug 2026 10:04:07 +0530
Subject: [PATCH] docs: add equations to inverse hyperbolic function READMEs

Add defining equations with logarithmic closed-form definitions to the
intro sections of asinh, acosh, atanh, acoth, asech, acsch and their
single-precision variants.

Ref: https://github.com/stdlib-js/metr-issue-tracker/issues/1257
---
 .../@stdlib/math/base/special/acosh/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/acoshf/README.md   | 16 ++++++++++++++++
 .../@stdlib/math/base/special/acoth/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/acothf/README.md   | 16 ++++++++++++++++
 .../@stdlib/math/base/special/acsch/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/acschf/README.md   | 16 ++++++++++++++++
 .../@stdlib/math/base/special/asech/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/asechf/README.md   | 16 ++++++++++++++++
 .../@stdlib/math/base/special/asinh/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/asinhf/README.md   | 16 ++++++++++++++++
 .../@stdlib/math/base/special/atanh/README.md    | 16 ++++++++++++++++
 .../@stdlib/math/base/special/atanhf/README.md   | 16 ++++++++++++++++
 12 files changed, 192 insertions(+)

diff --git a/lib/node_modules/@stdlib/math/base/special/acosh/README.md b/lib/node_modules/@stdlib/math/base/special/acosh/README.md
index 6747732f5427..1cb6ac728e2c 100644
--- a/lib/node_modules/@stdlib/math/base/special/acosh/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acosh/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arccosine][hyperbolic-arccosine] of a double-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arccosine][hyperbolic-arccosine] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccosine" align="center" raw="y = \operatorname{arccosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)" alt="Hyperbolic arccosine function"> -->
+
+```math
+y = \mathop{\mathrm{arccosh}}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/acoshf/README.md b/lib/node_modules/@stdlib/math/base/special/acoshf/README.md
index 7e6a41cc6684..bad71768f25e 100644
--- a/lib/node_modules/@stdlib/math/base/special/acoshf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acoshf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arccosine][hyperbolic-arccosine] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arccosine][hyperbolic-arccosine] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccosine" align="center" raw="y = \operatorname{arccosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)" alt="Hyperbolic arccosine function"> -->
+
+```math
+y = \mathop{\mathrm{arccosh}}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/acoth/README.md b/lib/node_modules/@stdlib/math/base/special/acoth/README.md
index 182cf05edece..8e64d43d5fe0 100644
--- a/lib/node_modules/@stdlib/math/base/special/acoth/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acoth/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [inverse hyperbolic cotangent][hyperbolic-arctangent] of a double-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arccotangent][hyperbolic-arctangent] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccotangent" align="center" raw="y = \operatorname{arccoth}(x) = \frac{1}{2} \ln\left(\frac{x+1}{x-1}\right)" alt="Hyperbolic arccotangent function"> -->
+
+```math
+y = \mathop{\mathrm{arccoth}}(x) = \frac{1}{2} \ln\left(\frac{x+1}{x-1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/acothf/README.md b/lib/node_modules/@stdlib/math/base/special/acothf/README.md
index 5532bc95223b..1ef010c8a6f9 100644
--- a/lib/node_modules/@stdlib/math/base/special/acothf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acothf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [inverse hyperbolic cotangent][hyperbolic-arctangent] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arccotangent][hyperbolic-arctangent] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccotangent" align="center" raw="y = \operatorname{arccoth}(x) = \frac{1}{2} \ln\left(\frac{x+1}{x-1}\right)" alt="Hyperbolic arccotangent function"> -->
+
+```math
+y = \mathop{\mathrm{arccoth}}(x) = \frac{1}{2} \ln\left(\frac{x+1}{x-1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/acsch/README.md b/lib/node_modules/@stdlib/math/base/special/acsch/README.md
index 55467efc620e..af1c60c80492 100644
--- a/lib/node_modules/@stdlib/math/base/special/acsch/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acsch/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arccosecant][inverse-hyperbolic-functions] of a number.
 
+<section class="intro">
+
+The [inverse hyperbolic cosecant][inverse-hyperbolic-functions] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccosecant" align="center" raw="y = \operatorname{arccsch}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)" alt="Hyperbolic arccosecant function"> -->
+
+```math
+y = \mathop{\mathrm{arccsch}}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/acschf/README.md b/lib/node_modules/@stdlib/math/base/special/acschf/README.md
index c585979d2cf3..dd33fff71998 100644
--- a/lib/node_modules/@stdlib/math/base/special/acschf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/acschf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arccosecant][inverse-hyperbolic-functions] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [inverse hyperbolic cosecant][inverse-hyperbolic-functions] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arccosecant" align="center" raw="y = \operatorname{arccsch}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)" alt="Hyperbolic arccosecant function"> -->
+
+```math
+y = \mathop{\mathrm{arccsch}}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/asech/README.md b/lib/node_modules/@stdlib/math/base/special/asech/README.md
index 496fc71f53c7..409f30e36d41 100644
--- a/lib/node_modules/@stdlib/math/base/special/asech/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/asech/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arcsecant][hyperbolic-arcsecant] of a number.
 
+<section class="intro">
+
+The [hyperbolic arcsecant][hyperbolic-arcsecant] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arcsecant" align="center" raw="y = \operatorname{arcsech}(x) = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)" alt="Hyperbolic arcsecant function"> -->
+
+```math
+y = \mathop{\mathrm{arcsech}}(x) = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/asechf/README.md b/lib/node_modules/@stdlib/math/base/special/asechf/README.md
index 5527253468ed..23b126555b05 100644
--- a/lib/node_modules/@stdlib/math/base/special/asechf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/asechf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arcsecant][hyperbolic-arcsecant] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arcsecant][hyperbolic-arcsecant] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arcsecant" align="center" raw="y = \operatorname{arcsech}(x) = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)" alt="Hyperbolic arcsecant function"> -->
+
+```math
+y = \mathop{\mathrm{arcsech}}(x) = \ln\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/asinh/README.md b/lib/node_modules/@stdlib/math/base/special/asinh/README.md
index 03f80748289c..394661421f00 100644
--- a/lib/node_modules/@stdlib/math/base/special/asinh/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/asinh/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arcsine][hyperbolic-arcsine] of a double-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arcsine][hyperbolic-arcsine] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arcsine" align="center" raw="y = \operatorname{arcsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)" alt="Hyperbolic arcsine function"> -->
+
+```math
+y = \mathop{\mathrm{arcsinh}}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/asinhf/README.md b/lib/node_modules/@stdlib/math/base/special/asinhf/README.md
index 3ded98fbe462..51e39a3279fc 100644
--- a/lib/node_modules/@stdlib/math/base/special/asinhf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/asinhf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arcsine][hyperbolic-arcsine] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arcsine][hyperbolic-arcsine] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arcsine" align="center" raw="y = \operatorname{arcsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)" alt="Hyperbolic arcsine function"> -->
+
+```math
+y = \mathop{\mathrm{arcsinh}}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/atanh/README.md b/lib/node_modules/@stdlib/math/base/special/atanh/README.md
index e4bd3cec47eb..933f5376410a 100644
--- a/lib/node_modules/@stdlib/math/base/special/atanh/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/atanh/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arctangent][hyperbolic-arctangent] of a double-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arctangent][hyperbolic-arctangent] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arctangent" align="center" raw="y = \operatorname{arctanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)" alt="Hyperbolic arctangent function"> -->
+
+```math
+y = \mathop{\mathrm{arctanh}}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage
diff --git a/lib/node_modules/@stdlib/math/base/special/atanhf/README.md b/lib/node_modules/@stdlib/math/base/special/atanhf/README.md
index 1117081fbef3..b9710878f967 100644
--- a/lib/node_modules/@stdlib/math/base/special/atanhf/README.md
+++ b/lib/node_modules/@stdlib/math/base/special/atanhf/README.md
@@ -22,6 +22,22 @@ limitations under the License.
 
 > Compute the [hyperbolic arctangent][hyperbolic-arctangent] of a single-precision floating-point number.
 
+<section class="intro">
+
+The [hyperbolic arctangent][hyperbolic-arctangent] function is defined as
+
+<!-- <equation class="equation" label="eq:hyperbolic_arctangent" align="center" raw="y = \operatorname{arctanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)" alt="Hyperbolic arctangent function"> -->
+
+```math
+y = \mathop{\mathrm{arctanh}}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage