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MathGroup Archive 2007

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Re: Strange results from Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75012] Re: Strange results from Mathematica
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sat, 14 Apr 2007 01:06:23 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <evn8cr$s7$1@smc.vnet.net>

Jung-Tsung Shen wrote:
> Hi, I encountered a rather strange results from Mathematica today.
> 
> When evaluating the following integration
> 
> (1) Integrate[x (BesselJ[0, x])^2, {x,0, infinity}]
==========================================^^^^^^^^^
Spelling error: infinity must be written Infinity (with a capital I)

> my Mathematica gave the answer 0. That's very weird, since if you plot
> out the function x (BesselJ[0, x])^2, apparently the integral should
> not be zero ...

You are right since this integral is divergent. Mathematica 5.2 for 
Microsoft Windows (June 20, 2005) returns

Integrate[x*BesselJ[0, x]^2, {x, 0, Infinity}]

  Integrate::"gener" : "Unable to check convergence. More...

  Integrate::"idiv" : "Integral of (x*\"BesselJ[0, x]\"^2) does not 
converge on ({0, Infinity}). More...

Integrate[x*BesselJ[0, x]^2, {x, 0, Infinity}]

You could try the following tho check the limit:

In[7]:=
Integrate[x*BesselJ[0, x]^2, {x, 0, t}]

Out[7]=
1  2               2
- t  (BesselJ[0, t]  +
2

                  2
     BesselJ[1, t] )

In[8]:=
Limit[%, t -> Infinity]

Out[8]=
Infinity

> Moreover, for the following integration
> 
> (2) Integrate[ (BesselJ[0, x])^2, {x,0, infinity}]
> 
> My Mathematica says it's some finite value. However, once you did
> this, and then did (1), and then came back to do (1), the answer
> became 0 ...

Not sure what you mean here. On my system, the expression is just 
returned unevaluated.

In[9]:=
Integrate[BesselJ[0, x]^2, {x, 0, Infinity}]

Out[9]=
Integrate[BesselJ[0, x]^2, {x, 0, Infinity}]

> My Mathematica version is 5.0.1.0 on Mac Powerbook.
> 
> Thanks.
> 
> JT
> 



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