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Re: Incorrect symbolic improper integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103715] Re: Incorrect symbolic improper integral
  • From: ADL <alberto.dilullo at tiscali.it>
  • Date: Sat, 3 Oct 2009 09:04:04 -0400 (EDT)
  • References: <200909301141.HAA14962@smc.vnet.net> <ha212q$n8u$1@smc.vnet.net>

Following Dan Dubin's comment, I made some tests, reported below with
some visual simplification, which show that there are some troubles in
assumptions management in Mathematica.
Note that, when I gave the parameter a specific value, real or
complex, I found no problems with this integral.

My version is 7.0.1 on Windows.
I do not think that DrMajorBob comments about Solve are applicable in
the cases below:

Assuming[a > 0, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> If[a > 1, (Pi*Csc[Pi/a])/a, (*otherwise...*)]
-OK-

Assuming[a < 0, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> Message: Integrate::idiv:Integral does not converge on {0,
Infinity}
==> Integrate[(1 + x^a)^(-1), {x, 0, Infinity}]
-OK-

Assuming[a != 0, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> If[Re[a] > 0, (Pi*Csc[Pi/a])/a, (*otherwise...*)]
-WRONG-

Assuming[Element[a, Reals], Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> If[a > 0, (Pi*Csc[Pi/a])/a, (*otherwise...*)]
-WRONG-

Assuming[a < 0, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> Message: Integrate::idiv:Integral does not converge on {0,
Infinity}
==> Integrate[(1 + x^a)^(-1), {x, 0, Infinity}]
-OK-

Assuming[a < 1, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> If[a > 0, (Pi*Csc[Pi/a])/a, (*otherwise...*)]
-WRONG-

Assuming[a < -1, Integrate[1/(1 + x^a), {x, 0, Infinity}]]
==> Message: Integrate::idiv:Integral does not converge on {0,
Infinity}
==> Integrate[(1 + x^a)^(-1), {x, 0, Infinity}]
-OK-


Regards
ADL

On Oct 1, 12:42 pm, Dan Dubin <ddu... at ucsd.edu> wrote:
> OK, many people have replied that the given integral was in fact done
> correctly by Mathematica. Here's a related integral that is not done
> correctly:
>
> Integrate[1/(1 + x^a),{x,0,Infinity}]
> ...
> This result is incorrect in the range 0<Re[a]<1. In this range the
> integral diverges, and is not given by the above cosecant expression.
> | Professor Dan Dubin
> | Dept of Physics , Mayer Hall Rm 3531,
> | UC San Diego La Jolla CA 92093-0319
> | phone (858) - 534-4174 fax: (858)-534-0173
> | ddu... at ucsd.edu



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