implement arbitrary precision Bessel Y function

[aghitza]

At the moment, Sage uses Maxima to compute the Bessel Y function. This is slow and works only with the default 53 bits of precision. It would be fairly easy to implement this:

• for integer values of the order nu, use the mpfr yn function
• for non-integer values of nu, use the formula $Y_nu(z) = (J_nu(z)cos(nupi) - J_{-nu}(z))/sin(nu*pi)$, where J is the Bessel J function.

CC: @kcrisman @benjaminfjones

Component: calculus

Reviewer: Karl-Dieter Crisman, Benjamin Jones

Issue created by migration from https://trac.sagemath.org/ticket/4230

[dandrake]

comment:1

It would also be nice to be able to evaluate Bessel functions with complex, or at least purely imaginary, arguments.

[aghitza]

comment:2

See #3426 (and review it!) for the Bessel functions other than Y. The code computes values at arbitrary complex coefficients.

[dandrake]

comment:3

Now that mpmath is included in Sage, why not just use mpmath's Bessel functions? http://mpmath.googlecode.com/svn/trunk/doc/build/functions/bessel.html

They seem to be very well-implemented, work to arbitrary precision, take complex arguments, and so on. Is this a good idea?

[kcrisman]

comment:5

This would most likely be fixed by #4102.

[benjaminfjones]

comment:6

Yep, I'll add a related doctest in #4102 to address arbitrary precision numerical evaluation for bessel_Y.

[kcrisman]

Reviewer: Karl-Dieter Crisman, Benjamin Jones

[kcrisman]

comment:7

Confirmed that this is doctested there.