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