# [Maxima] As x -> 0

Richard Hennessy rvh2007 at comcast.net
Sat Jul 5 12:48:14 CDT 2008

```I guess Wikipedia is wrong about this one.

(%i1) assume(a>0);
(%o1) [a > 2]
(%i2) limit(bessel_y(a,x)/((2/x)^a*-2/%pi*gamma(a-2)*(a-1)*(a-2)/2), x, 0);
The number 0 isn't in the domain of bessel_y -- an error.
To debug this try debugmode(true);

should be
(%o2) 1

based on this empirical data in a for loop in Maxima

for i : .1 thru 10 step .1123254 do print(ev(bessel_y(a,x)/((2/x)^a*-2/%pi*gamma(a-2)*(a-1)*(a-2)/2),x=.000000000000001,a=i,numer));

you get all ones (almost).  Wikipedia has the wrong formula, definitely.  But you have to watch out for a = 2 or a = 1 exactly,

Rich

------------Original Message------------
From: "Richard Hennessy"<rvh2007 at comcast.net>
To: "Maxima List" <maxima at math.utexas.edu>
Date: Sat, Jul-5-2008 10:53 AM
Subject: [Maxima] As x -> 0

I think these 2 limits should be 1 not a noun form and the questions are unnecessary.  My wish list is for the bessel functions to work this way.  Also the ones approaching infinity should be added to my wish list.

(%i1) assume(a>0);
(%o1) [a>0]
.
.
.

(%i7) limit(bessel_i(a,x)/(1/gamma(a+1)*(x/2)^a),x,0);
Is  bessel_i(a,0)*gamma(a+1)  positive, negative, or zero?zero;
Is  a-1  positive, negative, or zero?pos;
Is a an integer?no;
(%o7) 2^a*gamma(a+1)*(limit(bessel_i(a,x)/x^a,x,0))

(%i8) limit(bessel_i(a,x)/(1/gamma(a+1)*(x/2)^a),x,0);
Is  bessel_i(a,0)*gamma(a+1)  positive, negative, or zero?zero;
Is  a-1  positive, negative, or zero?neg;
Is a an integer?no;
(%o8) 2^a*gamma(a+1)*(limit(bessel_i(a,x)/x^a,x,0))

Rich

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