Subject: derivatives of bessel_j A&S 9.1.27 vs 9.1.30
From: Barton Willis
Date: Wed, 24 Jan 2007 17:04:41 -0600
maxima-bounces at math.utexas.edu wrote on 01/24/2007 02:54:42 PM:
> >>>>> "Robert" == Robert Dodier <robert.dodier at gmail.com> writes:
>
> Robert> On 1/24/07, Barton Willis <willisb at unk.edu> wrote:
> >> Would it be OK to change the derivatives of the Bessel functions
> >> to use A&S 9.1.27 instead of 9.1.30?
>
> Robert> OK by me.
>
> Is this related to the bug Barton reported about taylor not being able
> to expand bessel functions?
Yes.
> If so, we should also tell limit about bessel functions. (I think that
was the issue.)
Sure, telling limit about bessel is a good thing, but I don't think that
will fix the taylor(bessel_j(0,x),x,0,4) bug. I think taylor evaluates
by substitution, not via limit. I tested the change from A&S 9.1.27 to
9.1.30:
(%i22) load("bessel.lisp")$
(%i23) taylor(bessel_j(0,x),x,0,5);
(%o23) 1-x^2/4+x^4/64+...
(%i24) taylor(bessel_j(5,x),x,0,5);
(%o24) x^5/3840+...
(%i25) taylor(bessel_j(5,x),x,1,5);
(%o25) bessel_j(5,1)+((bessel_j(4,1)-bessel_j(6,1))*(x-1))/2 + junk
That's OK. But the same change for beseel_y is a problem:
(%i28) taylor(bessel_y(5,x),x,0,1);
(%o28) bessel_y(5,0)+((bessel_y(4,0)-bessel_y(6,0))*x)/2+...
I'm not sure how to handle that... yeechs:
(%i32) bessel_y(0,0);
(%o32) bessel_y(0,0) <--- could be an error or
> But otherwise, I have no problems with changing the derivatives.
> Perhaps the derivative in 9.1.30 makes it easier for maxima to deduce
> that the bessel functions satisfy Bessel's differential equation?
No, I don't think it makes it easier, just different.
Barton