• To: mathgroup at smc.vnet.net
• Subject: [mg89395] A question about N[...]
• From: "wyelen at gmail.com" <wyelen at gmail.com>
• Date: Sun, 8 Jun 2008 02:32:28 -0400 (EDT)

```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.