Add complex number support to sinh

.gitignore

@@ -30,3 +30,4 @@ node_modules/
 __pycache__/
 *.pyc
 spec/**/generated
+tmp/

spec/API_specification/array_api/elementwise_functions.py

@@ -1236,28 +1236,59 @@ def sin(x: array, /) -> array:
     """
 
 def sinh(x: array, /) -> array:
-    """
-    Calculates an implementation-dependent approximation to the hyperbolic sine, having domain ``[-infinity, +infinity]`` and codomain ``[-infinity, +infinity]``, for each element ``x_i`` of the input array ``x``.
+    r"""
+    Calculates an implementation-dependent approximation to the hyperbolic sine for each element ``x_i`` of the input array ``x``.
+
+    The mathematical definition of the hyperbolic sine is
+
+    .. math::
+       \operatorname{sinh}(x) = \frac{e^x - e^{-x}}{2}
 
     **Special cases**
 
-    For floating-point operands,
+    .. note::
+       For all operands, ``sinh(x)`` must equal ``-sinh(-x)``.
+
+    For real-valued floating-point operands,
 
     - If ``x_i`` is ``NaN``, the result is ``NaN``.
     - If ``x_i`` is ``+0``, the result is ``+0``.
     - If ``x_i`` is ``-0``, the result is ``-0``.
     - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
     - If ``x_i`` is ``-infinity``, the result is ``-infinity``.
 
+    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and
+
+    .. note::
+       For complex floating-point operands, ``sinh(conj(x))`` must equal ``conj(sinh(x))``.
+
+    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
+    - If ``a`` is ``+0`` and ``b`` is ``+infinity``, the result is ``0 + NaN j`` (sign of the real component is unspecified).
+    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``0 + NaN j`` (sign of the real component is unspecified).
+    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
+    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
+    - If ``a`` is ``+infinity`` and ``b`` is ``+0``, the result is ``+infinity + 0j``.
+    - If ``a`` is ``+infinity`` and ``b`` is a positive finite number, the result is ``+infinity * cis(b)``.
+    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``infinity + NaN j`` (sign of the real component is unspecified).
+    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``infinity + NaN j`` (sign of the real component is unspecified).
+    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
+    - If ``a`` is ``NaN`` and ``b`` is a nonzero finite number, the result is ``NaN + NaN j``.
+    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
+
+    where ``cis(v)`` is ``cos(v) + sin(v)*1j``.
+
+    .. note::
+       The hyperbolic sine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period :math:`2\pi j`, with respect to the imaginary component.
+
     Parameters
     ----------
     x: array
-        input array whose elements each represent a hyperbolic angle. Should have a real-valued floating-point data type.
+        input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
 
     Returns
     -------
     out: array
-        an array containing the hyperbolic sine 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 hyperbolic sine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.
     """
 
 def square(x: array, /) -> array: