Symbolic elliptic integrals

[eviatarbach]

This ticket is for making the elliptic integrals symbolic, including arbitrary-precision numeric evaluation.

Defect because elliptic_e(2.5,2.5).n() throwing an ECL error is fixed with it.

[eviatarbach]

Need to get full doctest coverage.

[rws]

Updated to 6.3beta6:

3 items had failures:
   1 of   4 in sage.functions.special.elliptic_j.EllipticEC
   1 of   2 in sage.functions.special.elliptic_j.EllipticEC._derivative_
   1 of   4 in sage.functions.special.elliptic_j.EllipticPi

---

New commits:

​a922acc

15046: Making elliptic integrals symbolic

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​67d51c9	15046: corrections to previous patch to pass doctests
​1b33207	15046: accept algorithm param for all evalf functions
​e950c33	15046: add all doctests; fix latex; augment documentation

[rws]

This is now ready for review.

BTW, is Maxima not able to solve the resp. integrals?

sage: integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)
sage: integrate(sqrt(1 - m*sin(x)^2), x, 0, pi/2)
sage: integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5547215

Merge branch 'develop' into t/15046/symbolic_elliptic_integrals

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​829da64

Merge branch 'develop' into t/15046/symbolic_elliptic_integrals

[jdemeyer]

Just to let you know: this conflicts with #17130.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d5c9eb7	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​ad02c9c	15046: simplifications due to BuiltinFunction improvements

[rws]

With this MaximaFunction is no longer used. I tend to put it in its own source file in functions. Comments?

[kcrisman]

With this MaximaFunction? is no longer used. I tend to put it in its own source file in functions.

Do you mean it isn't used at all? Wow! I guess that is a good thing, though.

[rws]

sage -t --long src/sage/functions/special.py  # 2 doctests failed

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1b5573b	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​e1e8900	15046: fix doctests

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​79c8fda

15046: add doctest for Maxima error

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f71294a	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​b3b2e8c	15046: fix doctest

[rws]

Passes all patchbot tests.

[kcrisman]

I'd like to look at this - can you confirm that you are giving positive review to Eviatar's original changes? If so, I think I only need to give this one more good look and try out some more numerical things to confirm it's ready to go.

By the way, I note some inconsistency (probably due to my relative unfamiliarity with elliptic integrals). I see

This returns the value of the "incomplete elliptic integral of the
    second kind"

in two places, EllipticE and EllipticEU. Since ​the Maxima manual clarifies the relationship, we should too. Indeed, there is a bunch of stuff in here which assumes familiarity with syntax for the Jacobi functions dn or sn or the rest which should be clarified. Maybe a link in internal documentation to that file, or perhaps some Wikipedia refs would also not be amiss? Just throwing out doc ideas, as long as we implement a little more it will be fine.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9a62ed2	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​27278e3	15046: add refs

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6138b32	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​fa4fd69	fix doctest

[rws]

Sage startup time increased by 0.3%, why?

[tscrim]

I am pretty sure that is from a bad startup time test batch. Also, could you break the longer lines to ~80 characters?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​bebff6c	Merge branch 'develop' into t/15046/symbolic_elliptic_integrals
​e4b089a	15046: cosmetics

[rws]

I didn't break all the lines. If you insist I'll try again. Anything else?

[tscrim]

I would prefer if the doctest output lines were split, probably 1 term per line should be good (or more if they are shorter). Once you do that, you can set a positive review on my behalf

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fc1ac57

15046: break doctest output lines

[rws]

I couldn't break lines at abc*xyz (bug?) but it worked at abc - xyz and abc + xyz so I succeeded in most of the cases.

[tscrim]

Thanks. What you did is exactly what I was looking for and plenty good. The reason why abc*xyz cannot split is because there is no whitespace around the operator (and newlines get counted as whitespace, which the doctest framework ignores all after one whitespace char).