Fix conversion from symbolic expressions with nonintegral exponents to power series ring

[TinaJin0228]

Fixes Issue #39736

This pull request addresses the issue where converting a symbolic expression with nonintegral exponents (e.g. sqrt(1+x)) into a power series leads to an error like:

ValueError: exponent must be a rational number

The problem arose because expressions such as (1+x)^(1/2) were being caught by the branch that handles instances already identified as power series, rather than being expanded via a Taylor series. In addition, a NameError was raised due to the missing import of the symbolic ring identifier (SR).

Changes Made:

Added a check to determine if the symbolic expression is not a polynomial (by testing using the series variable).

If the expression isn’t a polynomial, it is now passed through the lazy expansion branch using LazyPowerSeriesRing.taylor, ensuring that nonintegral exponents are properly expanded.

Import Fix:

Added an import for SR (using from sage.all import SR) so that SR.var(self.variable_name()) is defined.

Docstring Update:

Updated the element_constructor docstring with examples showing the new behavior for converting non-polynomial symbolic expressions.

Testing:

sage: R.<x> = PowerSeriesRing(QQ, default_prec=7)
sage: f = R(sqrt(1+x))
sage: f
1 + 1/2*x - 1/8*x^2 + 1/16*x^3 - 5/128*x^4 + 7/256*x^5 - 21/1024*x^6 + O(x^7)

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[user202729]

in which situation does this exception happen?

[TinaJin0228]

Considering no derivative method or singularity.

[user202729]

what does that mean? Can you add a doctest in which an exception is raised?

[TinaJin0228]

I didn't find examples where an exception was raised, so I deleted it. And I added some other doctests (for e^x and sin(x)) in the new commit.

[user202729]

Looks reasonable.

There's this issue

sage: var("x y")
(x, y)
sage: Frac(QQ[y])[[x]](y+1/(x+1))
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
...
AttributeError: 'PolynomialRing_field_with_category' object has no attribute '_SageObject__custom_na
me'

During handling of the above exception, another exception occurred:

TypeError                                 Traceback (most recent call last)
...
TypeError: cannot convert (-1)/1 to an element of Fraction Field of Univariate Polynomial Ring in y 
over Rational Field

Is this caused by this pull request? (or is this underlying bug with conversion to LazyPowerSeriesRing?)

[user202729]

what does this comment mean here (I thought we're not testing for exception here…?)

[TinaJin0228]

my fault, I forget to delete it after the testing...

[user202729]

There's another way to get the Taylor expansion

sage: (1/(x+1)).series(x, 10)
1 + (-1)*x + 1*x^2 + (-1)*x^3 + 1*x^4 + (-1)*x^5 + 1*x^6 + (-1)*x^7 + 1*x^8 + (-1)*x^9 + Order(x^10)

Are they equivalent, if not is there any significant difference (e.g. which one is faster?)