From 5032ce141f12f6e0ab326a3fb6ea24e8ea3097ec Mon Sep 17 00:00:00 2001
From: Athan Reines <kgryte@gmail.com>
Date: Mon, 5 Dec 2022 01:44:58 -0800
Subject: [PATCH 1/2] Update copy and add TODO

---
 spec/API_specification/array_api/elementwise_functions.py | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/spec/API_specification/array_api/elementwise_functions.py b/spec/API_specification/array_api/elementwise_functions.py
index e76be8d65..0ab01338f 100644
--- a/spec/API_specification/array_api/elementwise_functions.py
+++ b/spec/API_specification/array_api/elementwise_functions.py
@@ -1356,14 +1356,20 @@ def sign(x: array, /) -> array:
 
     **Special cases**
 
+    For real-valued operands,
+
     - If ``x_i`` is less than ``0``, the result is ``-1``.
     - If ``x_i`` is either ``-0`` or ``+0``, the result is ``0``.
     - If ``x_i`` is greater than ``0``, the result is ``+1``.
 
+    For complex floating-point operands,
+
+    TODO
+
     Parameters
     ----------
     x: array
-        input array. Should have a real-valued data type.
+        input array. Should have a numeric data type.
 
     Returns
     -------

From 914e33d92314e08a14f9d10a5bc02c0e37afea24 Mon Sep 17 00:00:00 2001
From: Athan Reines <kgryte@gmail.com>
Date: Sun, 11 Dec 2022 23:28:13 -0800
Subject: [PATCH 2/2] Add equations and special cases

---
 .../array_api/elementwise_functions.py        | 19 ++++++++++++++++---
 1 file changed, 16 insertions(+), 3 deletions(-)

diff --git a/spec/API_specification/array_api/elementwise_functions.py b/spec/API_specification/array_api/elementwise_functions.py
index 0ab01338f..34484162d 100644
--- a/spec/API_specification/array_api/elementwise_functions.py
+++ b/spec/API_specification/array_api/elementwise_functions.py
@@ -1351,9 +1351,19 @@ def round(x: array, /) -> array:
     """
 
 def sign(x: array, /) -> array:
-    """
+    r"""
     Returns an indication of the sign of a number for each element ``x_i`` of the input array ``x``.
 
+    The sign function (also known as the **signum function**) of a number :math:`x_i` is defined as
+
+    .. math::
+       \operatorname{sign}(x_i) = \begin{cases}
+       0 & \textrm{if } x_i = 0 \\
+       \frac{x}{|x|} & \textrm{otherwise}
+       \end{cases}
+
+    where :math:`|x_i|` is the absolute value of :math:`x_i`.
+
     **Special cases**
 
     For real-valued operands,
@@ -1361,10 +1371,13 @@ def sign(x: array, /) -> array:
     - If ``x_i`` is less than ``0``, the result is ``-1``.
     - If ``x_i`` is either ``-0`` or ``+0``, the result is ``0``.
     - If ``x_i`` is greater than ``0``, the result is ``+1``.
+    - If ``x_i`` is ``NaN``, the result is ``NaN``.
 
-    For complex floating-point operands,
+    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and
 
-    TODO
+    - If ``a`` is either ``-0`` or ``+0`` and ``b`` is either ``-0`` or ``+0``, the result is ``0 + 0j``.
+    - If ``a`` is ``NaN`` or ``b`` is ``NaN``, the result is ``NaN + NaN j``.
+    - In the remaining cases, special cases must be handled according to the rules of complex number division (see :func:`~array_api.divide`).
 
     Parameters
     ----------