• To: mathgroup at smc.vnet.net
• Subject: [mg89410] Re: A question about N[...]
• From: "Steve Luttrell" <steve at _removemefirst_luttrell.org.uk>
• Date: Mon, 9 Jun 2008 02:26:04 -0400 (EDT)
• References: <g2fug6\$2kg\$1@smc.vnet.net>

```This looks like a bug in HypergeometricPFQ.

To see this evaluate

Integrate[BesselJ[0, x r]^2, {r, 0, 20}]

to obtain

20 HypergeometricPFQ[{1/2,1/2},{1,1,3/2},-400 x^2]

and then plot this for 0<=x<=3 (i.e. including your required x=2.405 in the
range of the plot):

Plot[20 HypergeometricPFQ[{1/2,1/2},{1,1,3/2},-400
x^2],{x,0,3},PlotRange->{0,All}]

The HypergeometricPFQ[{1/2,1/2},{1,1,3/2},-400 x^2] evaluates correctly up
to around x=2.2, but is zero (i.e. wrong) for larger values of x.

Stephen Luttrell
West Malvern, UK

<wyelen at gmail.com> wrote in message news:g2fug6\$2kg\$1 at smc.vnet.net...
>
> Recently I came across a puzzling problem which I believed to be
> related to the function N.
>
> My platform is Mathematica 6.0 for Microsoft Windows (32-bit). When
> calculating the following
> integral, I got different results from Integrate & NIntegrate:
>
>          In[1]:= Integrate[BesselJ[0, 2.405 * r]^2, {r, 0, 20}]
>
>          Out[1]= 0.
>
>          In[2]:= NIntegrate[BesselJ[0, 2.405 * r]^2, {r, 0, 20}]
>
>          Out[2]= 0.864755
>
> Guessing a problem caused by numerical number 2.405, I rewrote it as
> an exact number:
>
>          In[3]:= Integrate[BesselJ[0, (2 + 405/1000)*r]^2, {r, 0,
> 20}]
>
>          Out[3]= 20*HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -
> (231361/100)]
>
> then evaluated the numerical value, which was surprisingly still 0.:
>
>          In[4]:= N[%]
>
>          Out[4]= 0.
>
> but evaluating with 6-digit precision gave the same result as
> NIntegrate:
>
>          In[5]:= N[%%,6]
>
>          Out[5]= 0.864755
>
> In help page for N it said "N[expr] is equivalent to
> N[expr,MachinePrecision]", but evaluating with a
> approximate precision didn't gave 0.:
>
>          In[6]:= N[MachinePrecision]
>
>          Out[6]= 15.9546
>
>          In[7]:= N[%3,15.9546]
>
>          Out[7]= 0.8647551857405188
>
> I wonder is this caused by the function N ,or whether I should just
> turn to another OS (say Linux) and things will go well.
>