Add complex support to acosh (#520)

Author: kgryte

spec/API_specification/array_api/elementwise_functions.py

@@ -51,27 +51,70 @@ def acos(x: array, /) -> array:
     """
 
 def acosh(x: array, /) -> array:
-    """
-    Calculates an implementation-dependent approximation to the inverse hyperbolic cosine, having domain ``[+1, +infinity]`` and codomain ``[+0, +infinity]``, for each element ``x_i`` of the input array ``x``.
+    r"""
+    Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element ``x_i`` of the input array ``x``.
 
     **Special cases**
 
-    For floating-point operands,
+    For real-valued floating-point operands,
 
     - If ``x_i`` is ``NaN``, the result is ``NaN``.
     - If ``x_i`` is less than ``1``, the result is ``NaN``.
     - If ``x_i`` is ``1``, the result is ``+0``.
     - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
 
+    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and
+
+    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``+0 + πj/2``.
+    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``+infinity + πj/2``.
+    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
+    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``NaN ± πj/2`` (sign of imaginary component is unspecified).
+    - If ``a`` is ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + πj``.
+    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + 0j``.
+    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + 3πj/4``.
+    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + πj/4``.
+    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
+    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
+    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``+infinity + NaN j``.
+    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
+
+    .. note::
+       The principal value of the inverse hyperbolic cosine of a complex number :math:`z` is
+
+       .. math::
+          \operatorname{acosh}(z) = \ln(z + \sqrt{z+1}\sqrt{z-1})
+
+       For any :math:`z`,
+
+       .. math::
+          \operatorname{acosh}(z) = \frac{\sqrt{z-1}}{\sqrt{1-z}}\operatorname{acos}(z)
+
+       or simply
+
+       .. math::
+          \operatorname{acosh}(z) = j\ \operatorname{acos}(z)
+
+       in the upper half of the complex plane.
+
+    .. note::
+       For complex floating-point operands, ``acosh(conj(x))`` must equal ``conj(acosh(x))``.
+
+    .. note::
+       The inverse hyperbolic cosine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segment :math:`(-\infty, 1)` of the real axis.
+
+       Accordingly, for complex arguments, the function returns the inverse hyperbolic cosine in the interval :math:`[0, \infty)` along the real axis and in the interval :math:`[-\pi j, +\pi j]` along the imaginary axis.
+
+       *Note: branch cuts have provisional status* (see :ref:`branch-cuts`).
+
     Parameters
     ----------
     x: array
-        input array whose elements each represent the area of a hyperbolic sector. Should have a real-valued floating-point data type.
+        input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
 
     Returns
     -------
     out: array
-        an array containing the inverse hyperbolic cosine of each element in ``x``. The returned array must have a real-valued floating-point data type determined by :ref:`type-promotion`.
+        an array containing the inverse hyperbolic cosine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.
     """
 
 def add(x1: array, x2: array, /) -> array: