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Re: large integration result for simple problem: 1/x,, also BesselJ
*To*: mathgroup at smc.vnet.net
*Subject*: [mg122862] Re: large integration result for simple problem: 1/x,, also BesselJ
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 12 Nov 2011 07:37:16 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201111110955.EAA08514@smc.vnet.net> <7999FCC4-7E77-4ED4-AEF6-AF3CCFAD0FA7@mimuw.edu.pl> <4EBD4CAB.1090505@eecs.berkeley.edu> <0CA89E70-9E70-4FC8-9E5D-4C16A4934A1F@mimuw.edu.pl> <4EBD5686.5030106@eecs.berkeley.edu>
On 11 Nov 2011, at 18:08, Richard Fateman wrote:
> On 11/11/2011 8:38 AM, Andrzej Kozlowski wrote:
>> Mathematica 8 returns:
>>
>>
>> Integrate[BesselJ[n, b*x], {x, 0, Infinity},
>> Assumptions -> {Re[n]> -1}]
>>
>> b^(n - 2)*(b^2)^(1/2 - n/2)
>>
>> Andrzej Kozlowski
>
> So by your previous note, this answer from version 8.0 is wrong since it does not exclude Im[b]==0.
> I note that the formula is also wrong unless it somehow excludes b==0, when the integral is infinite,
> but the formula is indeterminate.
I assume you must mean that it is wrong because it does not exclude Im[b]!=0 (in other words, a non-real number). If Im[b]!=0, e.g. b=I then
Integrate[BesselJ[n, I*x], {x, 0, Infinity}, Assumptions -> {Re[n] > -1}]
During evaluation of In[43]:= Integrate::idiv:Integral of Subscript[J, n](I x) does not converge on {0,\[Infinity]}. >>
Integrate[BesselJ[n, I*x], {x, 0, Infinity},
Assumptions -> {Re[n] > -1}]
b^(n - 2)*(b^2)^(1/2 - n/2) /. b -> I
(-1)^(1/2 - n/2)*I^(n - 2)
AK
>
> Interestingly Gradshteyn & Rhyzik exclude all b<=0 from their formula, with answer 1/b.
>
> G&R probably figure that a human would know about the symmetries of Bessel functions, and would deal with a
> negative coefficient in a sensible way. Just as an integral from -Infinity to 0 could be figured out, or some other integrals by a change of variables.
>
>
> RJF
>
>
>
> ..snip..
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