Add Hankel functions and make spherical Bessel and Hankel functions symbolic

[eviatarbach]

For some reason Sage has spherical Hankel functions but not regular ones. This patch adds Hankel functions and makes spherical Hankel and Bessel functions symbolic, as well as adding arbitrary-precision numeric evaluation.

Depends on #18257

CC: @burcin @fchapoton @paulmasson

Component: symbolics

Author: Eviatar Bach, Ralf Stephan

Branch/Commit: 7e48705

Reviewer: Paul Masson

Issue created by migration from https://trac.sagemath.org/ticket/15024

[eviatarbach]

Attachment: trac15024.patch.gz

[eviatarbach]

comment:1

Note that this also changes the LaTeX representation for the other Bessel functions by removing \operatorname, since in all sources I've seen the function name is italicized.

Patchbot apply trac15024.patch

[eviatarbach]

comment:2

New patch gets coverage to 100%, adds _latex_ methods (for getting a LaTeX representation for the function without arguments), and adds \left and \right to _print_latex_, which I forgot before.

Patchbot apply trac15024_2.patch

[sagetrac-vbraun-spam]

comment:3

Attachment: trac15024_2.patch.gz

[kcrisman]

comment:4

Note also the Struve functions

struve_h (v,z)                 Struve H function
struve_l (v,z)                 Struve L function

noted at the symbolics wiki on Trac, which seem to go with the Hankels...

[eviatarbach]

comment:5

Yeah, those should be easy to add, although maybe in another ticket.

[kcrisman]

comment:6

Sure, this is #16221 now. By the way, do we really want the Hankel stuff in bessel.py?

[rwst]

Branch: u/rws/add_hankel_functions_and_make_spherical_bessel_and_hankel_functions_symbolic

[rwst]

comment:9

Made commit from patch, removed merge conflicts. However:

sage -t --long src/sage/functions/special.py  # 2 doctests failed
sage -t --long src/sage/functions/bessel.py  # 12 doctests failed

---

New commits:

d0ada4b

Adding Hankel functions, making spherical Bessel and Hankel functions symbolic.

[rwst]

Commit: d0ada4b

[sagetrac-git]

Changed commit from d0ada4b to c2cd65d

[sagetrac-git]

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

57aa3d9	Merge branch 'develop' into t/15024/add_hankel_functions_and_make_spherical_bessel_and_hankel_functions_symbolic
c2cd65d	15024: fix doctests

[rwst]

comment:11

Please review. As to which category Hankel functions are in, if not "bessel", I would propose this hierarchy for orientation: special functions--->holonomic functions--->Bessel functions

[rwst]

Author: Eviatar Bach, Ralf Stephan

[sagetrac-git]

Changed commit from c2cd65d to 0e9578c

[sagetrac-git]

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

0e9578c

Merge branch 'develop' into t/15024/add_hankel_functions_and_make_spherical_bessel_and_hankel_functions_symbolic

[sagetrac-git]

Changed commit from 6f3fbbc to 2a5c206

[paulmasson]

comment:45

When running doctests I get the message

ImportError: cannot import name meval

Similar error when trying to build documentation. Sage crashes on startup, with message in crash log:

---> 56 from sage.functions.special import meval
...
ImportError: cannot import name meval

[paulmasson]

Reviewer: Paul Masson

[sagetrac-git]

Changed commit from 2a5c206 to fa0d0b0

[sagetrac-git]

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

fa0d0b0

15024: fix previous commit

[rwst]

comment:48

Oops, the change in airy.pyx is necessary because meval no longer exists.

[paulmasson]

comment:49

Doctests all pass. Documentation builds and has multiple preexisting issues, but that should be on a separate ticket.

Random numeric testing of new Hankel functions is accurate. Both functions plot. Symbolics behave as expected.

Per #20496 we are now supposed to escape abbreviated first names in references. There are three instances in the current branch that don't do that. If you fix those then I think it's ready to go.

There are changes in here I was planning to make, so I'm glad I read your list of open symbolics tickets beforehand!

[sagetrac-git]

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

821c307

cosmetics

[sagetrac-git]

Changed commit from fa0d0b0 to 821c307

[rwst]

comment:51

Thanks for the review. It's good to see again someone is interested in special functions. I am motivated in having at least the most known holonomic functions implemented symbolically (for experimental symbolics) but had stopped because very few tickets were reviewed.

---

New commits:

821c307

cosmetics

[paulmasson]

comment:52

Positive review after cosmetic changes but there is now a merge conflict with 7.3.beta7.

Do I need to review again after conflict is resolved? What's the protocol on this?

[sagetrac-git]

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

ab6006d

Merge branch 'develop' into t/15024/add_hankel_functions_and_make_spherical_bessel_and_hankel_functions_symbolic

[sagetrac-git]

Changed commit from 821c307 to ab6006d

[rwst]

comment:54

Replying to @paulmasson:

Do I need to review again after conflict is resolved? What's the protocol on this?

In principle I could have introduced any possible change with it, so yes. But it is easy to check this, either per eyeball by skimming the diff again, or automatically (I just tried this the first time ever):

git diff develop >diff1
git co 821c307        (the commit before the merge)
git diff 7.3.beta6 >diff2
diff -u diff2 diff1

The output of the last diff should not contain lines that start with ++/+-/-+/--. This way I found I somehow introduced some orig files into the branch so I'll delete those files. But as introducing and removing these files create completely unnecessary data I'll upload a new branch with an improved merge commit.

[rwst]

Changed branch from u/rws/add_hankel_functions_and_make_spherical_bessel_and_hankel_functions_symbolic to u/rws/15024

[paulmasson]

Changed commit from ab6006d to 2860630

[paulmasson]

comment:56

Doctests pass and documentation builds. Numerics look good. A few problems resulting from the merge conflict resolution:

1. A line has been modified in hypergeometric.py. This modification appears to be inaccurate and has the wrong variable anyway. Please remove the condition x>0 on hypergeometric_U.

2. The definition of the spherical Bessel function of the second kind has been left behind in special.py after the definition of spherical harmonics. This one needs to be deleted since it's already in bessel.py.

3. The evaluation of spherical harmonics has changed. I'm assuming this is because meval is gone, but please confirm. This change has no apparent problems.

4. There are doctests added to the complete elliptic integral which are outside this ticket. Since they pass I'm fine leaving them here, but let me know that they are included just to avoid an extra ticket.

---

New commits:

2860630

Merge branch 'develop' into tmp02

[sagetrac-git]

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

7e48705

15024: more cosmetics

[sagetrac-git]

Changed commit from 2860630 to 7e48705

[rwst]

comment:58

Replying to @paulmasson:

3. The evaluation of spherical harmonics has changed. I'm assuming this is because meval is gone, but please confirm.

Yes.

4. There are doctests added to the complete elliptic integral which are outside this ticket. Since they pass I'm fine leaving them here, but let me know that they are included just to avoid an extra ticket.

I cannot avoid the extra ticket but would ask you to leave this part in for now.

[paulmasson]

comment:59

Good to go.

[rwst]

comment:60

Thanks for your patient work.

[vbraun]

Changed branch from u/rws/15024 to 7e48705