Re: Re: Incorrect symbolic improper integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg103661] Re: [mg103636] Re: Incorrect symbolic improper integral*From*: Dan Dubin <ddubin at ucsd.edu>*Date*: Thu, 1 Oct 2009 06:41:53 -0400 (EDT)*References*: <200909301141.HAA14962@smc.vnet.net>

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}] The result given in v. 7.0.1 is If[Re[a] > 0, (\[Pi] Csc[\[Pi]/a])/a, Integrate[1/(1 + x^a), {x, 0, \[Infinity]}, Assumptions -> Re[a] <= 0]] 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. >The integral you tried is a classical one. It is always calculated >in the textbooks on application of >complex variables to calculation of integrals. Its exact value is >therefore, well-known. Evaluate this please: > >HoldForm[\!\( >\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \ >\(\[Infinity]\)]\( >FractionBox[\(Cos[a\ x]\), \( >\*SuperscriptBox[\(b\), \(2\)] + >\*SuperscriptBox[\(x\), \(2\)]\)] \[DifferentialD]x\)\) = \[Pi]/ > b Exp[-a b]] > >assuming a>0 and b>0. Its evaluation at a=b=1 yields: > >In[5]:= \[Pi]/b Exp[-a b] /. {a -> 1, b -> 1} > >Out[5]= \[Pi]/\[ExponentialE] > >which is obviously the same as the solution returned by Mathematica >that you showed. Another point that when I evaluated your >integral with parameter > >In[6]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}, > Assumptions -> a \[Element] Reals] > > >Out[6]= \[ExponentialE]^-Abs[a] \[Pi] > > it returned (by my machine Math 6.0, Windows XP) > >\[ExponentialE]^-Abs[a] \[Pi] > >that is in line with the above exact solution, rather than with the value > >\[Pi] Cosh[a] > >that you report. So may be you still have a problem. > >Alexei > > >Below is a definite integral that Mathematica does incorrectly. >Thought someone might like to know: > >In[62]:= Integrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}] > >Out[62]= \[Pi]/E > >What a pretty result--if it were true. The correct answer is \[Pi]*Cosh >[1], which can be checked by adding a new parameter inside the >argument of Cos and setting it to 1 at the end: > >In[61]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}, > Assumptions -> a \[Element] Reals] > >Out[61]= \[Pi] Cosh[a] > >Regards, > >Jason Merrill > >-- >Alexei Boulbitch, Dr., habil. >Senior Scientist > >IEE S.A. >ZAE Weiergewan >11, rue Edmond Reuter >L-5326 Contern >Luxembourg > >Phone: +352 2454 2566 >Fax: +352 2454 3566 > >Website: www.iee.lu > >This e-mail may contain trade secrets or privileged, undisclosed or >otherwise confidential information. If you are not the intended >recipient and have received this e-mail in error, you are hereby >notified that any review, copying or distribution of it is strictly >prohibited. Please inform us immediately and destroy the original >transmittal from your system. Thank you for your co-operation. -- --------------- | 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 | ddubin at ucsd.edu