Split the monstrous C API manual files in smaller parts.

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+.. highlightlang:: c
+
+.. _complexobjects:
+
+Complex Number Objects
+----------------------
+
+.. index:: object: complex number
+
+Python's complex number objects are implemented as two distinct types when
+viewed from the C API:  one is the Python object exposed to Python programs, and
+the other is a C structure which represents the actual complex number value.
+The API provides functions for working with both.
+
+
+Complex Numbers as C Structures
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Note that the functions which accept these structures as parameters and return
+them as results do so *by value* rather than dereferencing them through
+pointers.  This is consistent throughout the API.
+
+
+.. ctype:: Py_complex
+
+   The C structure which corresponds to the value portion of a Python complex
+   number object.  Most of the functions for dealing with complex number objects
+   use structures of this type as input or output values, as appropriate.  It is
+   defined as::
+
+      typedef struct {
+         double real;
+         double imag;
+      } Py_complex;
+
+
+.. cfunction:: Py_complex _Py_c_sum(Py_complex left, Py_complex right)
+
+   Return the sum of two complex numbers, using the C :ctype:`Py_complex`
+   representation.
+
+
+.. cfunction:: Py_complex _Py_c_diff(Py_complex left, Py_complex right)
+
+   Return the difference between two complex numbers, using the C
+   :ctype:`Py_complex` representation.
+
+
+.. cfunction:: Py_complex _Py_c_neg(Py_complex complex)
+
+   Return the negation of the complex number *complex*, using the C
+   :ctype:`Py_complex` representation.
+
+
+.. cfunction:: Py_complex _Py_c_prod(Py_complex left, Py_complex right)
+
+   Return the product of two complex numbers, using the C :ctype:`Py_complex`
+   representation.
+
+
+.. cfunction:: Py_complex _Py_c_quot(Py_complex dividend, Py_complex divisor)
+
+   Return the quotient of two complex numbers, using the C :ctype:`Py_complex`
+   representation.
+
+
+.. cfunction:: Py_complex _Py_c_pow(Py_complex num, Py_complex exp)
+
+   Return the exponentiation of *num* by *exp*, using the C :ctype:`Py_complex`
+   representation.
+
+
+Complex Numbers as Python Objects
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+
+.. ctype:: PyComplexObject
+
+   This subtype of :ctype:`PyObject` represents a Python complex number object.
+
+
+.. cvar:: PyTypeObject PyComplex_Type
+
+   This instance of :ctype:`PyTypeObject` represents the Python complex number
+   type. It is the same object as ``complex`` and ``types.ComplexType``.
+
+
+.. cfunction:: int PyComplex_Check(PyObject *p)
+
+   Return true if its argument is a :ctype:`PyComplexObject` or a subtype of
+   :ctype:`PyComplexObject`.
+
+   .. versionchanged:: 2.2
+      Allowed subtypes to be accepted.
+
+
+.. cfunction:: int PyComplex_CheckExact(PyObject *p)
+
+   Return true if its argument is a :ctype:`PyComplexObject`, but not a subtype of
+   :ctype:`PyComplexObject`.
+
+   .. versionadded:: 2.2
+
+
+.. cfunction:: PyObject* PyComplex_FromCComplex(Py_complex v)
+
+   Create a new Python complex number object from a C :ctype:`Py_complex` value.
+
+
+.. cfunction:: PyObject* PyComplex_FromDoubles(double real, double imag)
+
+   Return a new :ctype:`PyComplexObject` object from *real* and *imag*.
+
+
+.. cfunction:: double PyComplex_RealAsDouble(PyObject *op)
+
+   Return the real part of *op* as a C :ctype:`double`.
+
+
+.. cfunction:: double PyComplex_ImagAsDouble(PyObject *op)
+
+   Return the imaginary part of *op* as a C :ctype:`double`.
+
+
+.. cfunction:: Py_complex PyComplex_AsCComplex(PyObject *op)
+
+   Return the :ctype:`Py_complex` value of the complex number *op*.
+
+   .. versionchanged:: 2.6
+      If *op* is not a Python complex number object but has a :meth:`__complex__`
+      method, this method will first be called to convert *op* to a Python complex
+      number object.