Make Airy functions symbolic

[sagetrac-olazo]

As discussed in sage-support.

Currently sage can evaluate airy functions numerically:

sage: airy_ai(1.4)
0.0820380498076

but it doesn't work symbolically

sage: airy_ai(x)
...
TypeError: Cannot evaluate symbolic expression to a numeric value.

We should make it symbolical for both airy_ai and airy_bi, as well as their derivatives.

Depends on #12289
Depends on #17130

CC: @kcrisman @burcin @benjaminfjones @fredrik-johansson @eviatarbach

Component: symbolics

Keywords: Airy functions sd40.5 sd48

Author: Oscar Gerardo Lazo Arjona, Benjamin Jones, Douglas McNeil, Eviatar Bach, Ralf Stephan

Branch: 2f6945a

Reviewer: Eviatar Bach, Karl-Dieter Crisman, Burcin Erocal, Ralf Stephan, Jeroen Demeyer, Marc Mezzarobba

Issue created by migration from https://trac.sagemath.org/ticket/12455

[sagetrac-olazo]

Description changed:

--- 
+++ 
@@ -1,3 +1,5 @@
+As discussed in [sage-support](http://groups.google.com/group/sage-support/browse_thread/thread/f458d0f9cfd89c9a).
+
 Currently sage can evaluate airy functions numerically:
 
 ```

[sagetrac-olazo]

comment:4

Attachment: trac_12455-symbolic_airy.patch.gz

I've added a patch, which should do the job, but it has a few shortcomings:

1.-The resulting symbolic functions seem to remain on hold:

sage: airy_ai(1.0)
airy_ai(1.00000000000000)

You need to force it to evaluate:

sage: airy_ai(1.0).n()
0.135292416313

2.- This doesn't work:

sage: airy_ai(2.0).n(digits=100)
0.0349241304233

3.- There is no evaluation for airy_ai_prime or airy_bi_prime

4.- I'm not sure about how should the functions be called, some possible schemes are

{ai,bi,aip,bip}
{ai,bai,aip,baip}
{airy_ai,airy_bi,airy_ai_prime,airy_bi_prime}

And also whether the latex representation should be capitalized or not. I chose the third scheme, and capitalized typesetting.

[kcrisman]

comment:6

Good start. A few responses.

• You'll need to have more methods. We should probably use mpmath to evaluate these functions in general, though using the SciPy version could be useful over RDF. See Sage is missing the lambert_w function #11888 for a very recent example of this kind of thing and how to implement it.
• That should also fix precision issues, if properly done, I think.
• SciPy has evaluation for the airy primes. I think that Maxima has these too. (Cc:ing FJ in case he has plans to add these to mpmath.)
• We should have translations of these to Maxima and friends. That will probably help us name them. What do Mathematica and Maple call them?
• All methods, including initialization methods, need doctests. Many of the new symbolic functions added at symbolics/functions will have nice models to follow.

Thanks for getting us a start on this! That's great.

[fredrik-johansson]

comment:7

mpmath has the derivatives (as well as integrals) -- just use the optional 'derivative' parameter.