https://docs.python.org/3/library/functions.html#pow

The built-in pow() function will return a complex number if the base is negative and the exponent is a float between 0 and 1. For example, the value returned by `pow(-1, 1.0/3)` is `(1.0000000000000002+1.7320508075688772j)`

The answer is mathematically correct, but `-2.0` is also mathematically correct. There is nothing in the documentation currently to suggest that a complex number might be returned; in fact, given the statement "[with] mixed operand types, the coercion rules for binary arithmetic operators apply", one might reasonably expect `-2.0` as the answer. 

I suggest the following sentences be added to the end of the second paragraph:

"If `base` is negative and the `exp` is a `float` between 0 and 1, a complex number will be returned. For example, `pow(-8, 1.0/3)` will return `(1.0000000000000002+1.7320508075688772j)`, and not `-2.0.`"

For some prior art, https://www.wolframalpha.com/input/?i=%28-8%29+%5E+%281%2F3%29 says it defaults to using "the principal root" over "the real-valued root"

Also, I think the relevant property is that the exponent is not an integer; being between 0 and 1 is irrelevant:

>>> pow(-8, 4/3)
(-8.000000000000005-13.856406460551014j)

Maybe the tweak could be something like

"Note that using a negative base with a non-integer exponent will return the principal complex exponent value, even if a different real value exists."

[Dennis]

> I think the relevant property is that the exponent is not an integer

Yep: the delegation to complex pow kicks in after handling infinities and nans, and only for strictly negative base (-0.0 doesn't count as negative for this purpose) and non-integral exponent.

Here's the relevant code: https://github.com/python/cpython/blob/257e400a19b34c7da6e2aa500d80b54e4c4dbf6f/Objects/floatobject.c#L773-L782

To avoid confusion, we should probably not mention fractions like `1/3` and `4/3` as example exponents in the documentation, since those hit the What-You-See-Is-Not-What-You-Get nature of binary floating-point. Mathematically, `z^(1/3)` is a very different thing from `z^(6004799503160661/18014398509481984)` for a negative real number `z`, and the latter is what's _actually_ being computed with `z**(1/3)`. The advantage of the principal branch approach is that it's continuous in the exponent, so that `z^(1/3)` and `z^(6004799503160661/18014398509481984)` only differ by a tiny amount.

I still think the most important aspect of this is that pow() will return complex numbers, contrary to what is implied by the statement I quoted at the beginning of this thread. 

Perhaps we should just borrow from the documentation for the power operator, which says:

Raising 0.0 to a negative power results in a ZeroDivisionError. Raising a negative number to a fractional power results in a complex number. (In earlier versions it raised a ValueError.)

https://docs.python.org/3/reference/expressions.html#the-power-operator

> Perhaps we should just borrow from the documentation for the power operator, which says [...]

That sounds good to me.

@Erik: Do you have a moment to look at the PR (GH-27853) and see if the proposed changes work for you?

The pow() docs could use substantial updating.  All of the logic for pow() is implemented in base.__pow__(exp, [mod]).  Technically, it isn't restricted to numeric types, that is just a norm.  The relationship between "pow(a,b)" and "a**b" is that both make the same call, a.__pow__(b).  The discussion of coercion rules dates back to Python 2 where used to have a coerce() builtin.  Now, the cross-type logic is buried in either in type(a).__pow__ or in type(b).__rpow__.  The equivalence of pow(a, b, c) to a more efficient form of "a ** b % c" is a norm but not a requirement and is only supported by ints or third-party types.

My suggestions

1st paragraphs:  Stay at a high level, covering only the most common use case and simple understanding of how most people use pow():

   Return *base* to the power *exp* giving the same result
   as ``base**exp``.

   If *mod* is present and all the arguments are integers,
   return *base* to the power *exp*, modulo *mod*.  This
   gives the same result as ``base ** exp % mod`` but is
   computed much more efficiently.

2nd paragraph:  Be precise about what pow() actually does, differentiating the typical case from what is actually required:

   The :func:`pow` function calls the base's meth:`__pow__` method
   falling back to the exp's meth:`__rpow__` if needed.  The logic
   and semantics of those methods varies depending on the type.
   Typically, the arguments have numeric types but this is not required.
   For types that support the three-argument form, the usual semantics
   are that ``pow(b, e, m)`` is equivalent to ``a ** b % c`` but
   this is not required.

3rd paragraph: Cover behaviors common to int, float, and complex.

4th paragraph and later:  Cover type specific behaviors (i.e. only int supports the three argument form and the other arguments must be ints as well).

New changeset 887a55705bb6c05a507c2886c9978a9e0cff0dd7 by Mark Dickinson in branch 'main':
bpo-44344: Document that pow can return a complex number for non-complex inputs. (GH-27853)
https://github.com/python/cpython/commit/887a55705bb6c05a507c2886c9978a9e0cff0dd7

New changeset 9b3cda56870d087cf50f605e91f3d26964868640 by Miss Islington (bot) in branch '3.10':
bpo-44344: Document that pow can return a complex number for non-complex inputs. (GH-27853) (GH-29135)
https://github.com/python/cpython/commit/9b3cda56870d087cf50f605e91f3d26964868640

New changeset c53428fe8980aab6eda3e573bafed657e6798e6e by Miss Islington (bot) in branch '3.9':
bpo-44344: Document that pow can return a complex number for non-complex inputs. (GH-27853) (GH-29134)
https://github.com/python/cpython/commit/c53428fe8980aab6eda3e573bafed657e6798e6e

Thanks, Mark! ✨ 🍰 ✨