Re: Problem with evaluation of Besel Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg55994] Re: Problem with evaluation of Besel Functions
• From: dh <dh at metrohm.ch>
• Date: Wed, 13 Apr 2005 01:10:22 -0400 (EDT)
• References: <d3g8kd\$smh\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi Ariel,
BesselJ[1.5,y]/.y->0 is only defined as a limit.
You will agree that 0/0 has no meaning. It can get one if we define it
as a limit where the numerator and denumerator tend towards zero with
certain rates.
Well, now consider: BesselJ[1.5,y]:

(0.7978845608028653558798921199`16.255619765854984*
(Cos[3.141592653589793 - y] +
Sin[3.141592653589793 - y]/y))/Sqrt[y]\
Times[Power[y, -1], Sin[Plus[3.141592653589793`, Times[-1, y]]]]]]

This is only meaningful in the limit sense. But for Mathematica this is
now an ordinary expression, no more BesselJ, and it is evaluated as
such, piece by piece. To see how it is evaluated you must look at the
internal form:
BesselJ[1.5, y] // FullForm

Times[0.7978845608028653558798921199`16.255619765854984, Power[y, \
Rational[-1, 2]], Plus[Cos[Plus[3.141592653589793`, Times[-1, y]]], \
Times[Power[y, -1], Sin[Plus[3.141592653589793`, Times[-1, y]]]]]]

The head of this expression is Time. First factor is a constant. The
second factor starts with:
Power[y, Rational[-1, 2]]
When this is evaluated for y==0 it naturally fails.

However, if you say BesselJ[1.5,0] Mathematica knows that it needs the
limit and you get the right answer.

After what I said you must get the right answer by:
Limit[BesselJ[3/2,y],y->0]  gives 0
what is true.

But here seems to lurck some nice pitfall:
Limit[BesselJ[1.5,y],y->0] gives Infinity.
Limit seems to have large problems with approximate Numbers. Can anybody
comment on this??

Sincerely, Daniel

Ariel sumeruk wrote:
> Hello
> I am having a problem evaluating various functions, One example is the following
>
> BesselJ[1.5, 0] evaluates to 0 but
> (BesselJ[xx, yyy] /. xx -> 1.5) /. yyy -> 0 evaluates to complex infinity
>
> I seem to encounter many of these problems with Bessel and Legendre
> functions where I get actual diffrent numerical results depending on
> How I set the parameters.
> Thanks for anyone who might help
> Ariel
>

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