Make elliptic_j function symbolic

[sagetrac-davidamadore]

Sage's elliptic_j function does not behave similarly to the other standard or special functions: it tries to numerically evaluate instead of building a symbolic expression:

sage: elliptic_j(I)
1728.00000000000
sage: elliptic_j(x)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
...
TypeError: Cannot evaluate symbolic expression to a numeric value.

This is very weird: it means, for example, that plot(real(elliptic_j(x+I)), (x,0,1)) will fail (instead, one must use plot(lambda t: real(elliptic_j(t+I)), (0,1)))

• http://mpmath.googlecode.com/svn/tags/0.15/doc/build/functions/elliptic.html#kleinj
• https://en.wikipedia.org/wiki/J-invariant

Component: symbolics

Keywords: special function, elliptic

Issue created by migration from https://trac.sagemath.org/ticket/15354

[rwst]

Changed keywords from special function to special function, elliptic

[grhkm21]

I looked at this just now, it doesn't seem too trivial, as the current implementation actually calls pari:

    CC = z.parent()
    if not isinstance(CC, sage.rings.abc.ComplexField):
        from sage.rings.complex_mpfr import ComplexField
        CC = ComplexField(prec)
        try:
            z = CC(z)
        except ValueError:
            raise ValueError("elliptic_j only defined for complex arguments.")
    from sage.libs.pari.all import pari
    return CC(pari(z).ellj())

I'm not sure how you can represent the function symbolically either, since it is transcendental in general. However, it will be nice for the function to be exact when the argument is an exact imaginary quadratic term. I will open an issue for that. Other than that, I will close the issue for now, if anyone has a better idea feel free to reopen.

[user202729]

@grhkm21 Old issue but it can certainly be represented symbolically. I guess the issue can be kept as a tracking issue.

For example with gamma:

sage: gamma(5)
24
sage: gamma(5, hold=True)
gamma(5)

(Though I don't understand enough about how Sage represents symbolic expression in general, I think the symbolic expression tree must be recognize-able by Maxima for things like e.g. simplify() to work? Through a cursory search Maxima doesn't seem to have modular form.)