Add complex number support to exp (#451)

* Update ignore file

* Add complex number support to `exp`

* Update input and output data types

* Update description to accommodate complex number input

* Add note concerning complex conjugation

* Specify special case in terms of `cis` function

* Fix special cases and move note

* Fix `cis` argument

Author: kgryte

.gitignore

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

spec/API_specification/array_api/elementwise_functions.py

@@ -223,8 +223,8 @@ def atan2(x1: array, x2: array, /) -> array:
     - If ``x1_i`` is greater than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``+π``.
     - If ``x1_i`` is less than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``+infinity``, the result is ``-0``.
     - If ``x1_i`` is less than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``-π``.
-    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is finite, the result is an implementation-dependent approximation to ``+π/2``.
-    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is finite, the result is an implementation-dependent approximation to ``-π/2``.
+    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a finite number, the result is an implementation-dependent approximation to ``+π/2``.
+    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a finite number, the result is an implementation-dependent approximation to ``-π/2``.
     - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``+infinity``, the result is an implementation-dependent approximation to ``+π/4``.
     - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``+3π/4``.
     - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``+infinity``, the result is an implementation-dependent approximation to ``-π/4``.
@@ -519,27 +519,51 @@ def equal(x1: array, x2: array, /) -> array:
 
 def exp(x: array, /) -> array:
     """
-    Calculates an implementation-dependent approximation to the exponential function, having domain ``[-infinity, +infinity]`` and codomain ``[+0, +infinity]``, for each element ``x_i`` of the input array ``x`` (``e`` raised to the power of ``x_i``, where ``e`` is the base of the natural logarithm).
+    Calculates an implementation-dependent approximation to the exponential function for each element ``x_i`` of the input array ``x`` (``e`` raised to the power of ``x_i``, where ``e`` is the base of the natural logarithm).
 
     **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 ``+0``, the result is ``1``.
     - If ``x_i`` is ``-0``, the result is ``1``.
     - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
     - If ``x_i`` is ``-infinity``, the result is ``+0``.
 
+    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and
+
+    .. note::
+       For complex floating-point operands, ``exp(conj(x))`` must equal ``conj(exp(x))``.
+
+    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``1 + 0j``.
+    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
+    - If ``a`` is a 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 finite number, the result is ``+0 * cis(b)``.
+    - If ``a`` is ``+infinity`` and ``b`` is a nonzero finite number, the result is ``+infinity * cis(b)``.
+    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``0 + 0j`` (signs of real and imaginary components are unspecified).
+    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
+    - If ``a`` is ``-infinity`` and ``b`` is ``NaN``, the result is ``0 + 0j`` (signs of real and imaginary components are unspecified).
+    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
+    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
+    - If ``a`` is ``NaN`` and ``b`` is not equal to ``0``, 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 exponential function is an entire function in the complex plane and has no branch cuts.
+
     Parameters
     ----------
     x: array
-        input array. Should have a real-valued floating-point data type.
+        input array. Should have a floating-point data type.
 
     Returns
     -------
     out: array
-        an array containing the evaluated exponential function result for 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 evaluated exponential function result for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.
     """
 
 def expm1(x: array, /) -> array: