make bessel_J symbolic

[jicama]

The motivation for this is

sage: plot(bessel_J(1, x), (x, 1, 10))
Traceback (most recent call last):
...
TypeError: Unable to convert x (='1-1/8*x^2+1/192*x^4-1/9216*x^6+1/737280*x^8-1/88473600*x^10+1/14863564800*x^12-1/3329438515200*x^14+1/958878292377600*x^16+O(x^17)') to real number.

The problem is that special functions, or at least bessel_J, can't currently be partially evaluated--that is, called with a SymbolicExpression as an argument. The model of good behavior is polylog, for which the above method produces a perfectly nice plot

sage: plot(polylog(1,x),(x,.1,.9)) #makes a fine plot

See discussion at http://groups.google.com/group/sage-support/browse_thread/thread/1b985b080ba2369e/7dee9eed953857f5#7dee9eed953857f5

---

Release manager:

Apply:

• attachment: trac_symbolic_bessel_v7.2.patch,
• attachment: trac_symbolic_bessel_v7-doctests.patch,
• attachment: bessel_2.2.patch
• attachment: trac_4102-bessel_doctest_fixes2.patch

Depends on #9880

CC: @jasongrout @kcrisman @benjaminfjones @sagetrac-dsm @burcin @eviatarbach

Component: calculus

Keywords: sd48

Author: Benjamin Jones, Eviatar Bach, Volker Braun

Reviewer: Karl-Dieter Crisman, Burcin Erocal

Merged: sage-5.11.rc0

Issue created by migration from https://trac.sagemath.org/ticket/4102

[dandrake]

comment:1

Orthogonal polynomials seem to work fine; you can plot the legendre_P, hermite, and jacobi_P polynomials with plot(legendre_P(4, x), (x,-1,1)) and so on. So whatever magic they are using might work for the Bessel functions and other special functions.

[jasongrout]

Changed keywords from none to jason

[jasongrout]

Changed keywords from jason to none

[burcin]

comment:5

This ticket description was too broad. We have lots of tickets on fixing symbolic functions, see symbolics/functions for an overview.

See #3426 and #4230 for other tickets related to bessel functions. It would make sense to fix all these together, with a base class that handles the common properties of bessel functions (differentiation), and subclasses that implement numerical evaluation, etc. for each type.

[benjaminfjones]

Description changed:

--- 
+++ 
@@ -10,7 +10,7 @@
 The problem is that special functions, or at least `bessel_J`, can't currently be partially evaluated--that is, called with a `SymbolicExpression` as an argument.  The model of good behavior is `polylog`, for which the above method produces a perfectly nice plot
 
 ```
-sage: polylog(1,x),(x,.1,.9) #makes a fine plot
+sage: plot(polylog(1,x),(x,.1,.9)) #makes a fine plot
 ```
 
 See discussion at http://groups.google.com/group/sage-support/browse_thread/thread/1b985b080ba2369e/7dee9eed953857f5#7dee9eed953857f5

[kcrisman]

comment:7

See also #10636, which I somehow never saw before. This sage-support thread yields an interesting related error:

sage: var('k')
k
sage: Z = sum( ((-1)^k*(x^(2*k+1))/factorial(2*k+1)),k,0,oo)
sage: Z
1/2*sqrt(pi)*sqrt(2)*sqrt(x)*bessel_j(1/2, x)
sage: Z(x=3)
1/2*sqrt(pi)*sqrt(2)*sqrt(3)*bessel_j(1/2, 3)
sage: Z(x=3).n()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

sage.symbolic.expression.Expression._numerical_approx (sage/symbolic/expression.cpp:20822)()

TypeError: cannot evaluate symbolic expression numerically
sage: besse
bessel_I  bessel_J  bessel_K  bessel_Y  

Note that we apparently aren't yet converting Maxima's Bessel properly to 'our' uppercase version because of this.

[benjaminfjones]

comment:8

This wouldn't be hard to implement using #11143 as a template, but what to do about names? I guess new names for the symbolic Bessel functions should be chosen and deprecation notices added to the existing Bessel_J etc.

What new names do people like in the interim?

• bessel_J_symbolic
• bessel_J_sym
• bessel_Js

???

[burcin]

comment:9

Replying to @benjaminfjones:

What new names do people like in the interim?

• bessel_J_symbolic
• bessel_J_sym
• bessel_Js

???

What is wrong with taking over bessel_{I,J,K,Y}? Why do we need interim names?

Thanks for looking into this.

[kcrisman]

comment:10

What is wrong with taking over bessel_{I,J,K,Y}? Why do we need interim names?

I concur.

• bessel_J_symbolic

One could use that as the "symbolic" class one and then do the usual foo = Foo_Class()?

Thanks for looking into this.

Agreed!