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Re: Improper integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37823] Re: [mg37801] Improper integral
  • From: Vladimir Bondarenko <vvb at mail.strace.net>
  • Date: Thu, 14 Nov 2002 06:11:26 -0500 (EST)
  • In-reply-to: <200211130611.BAA12200@smc.vnet.net>
  • References: <200211130611.BAA12200@smc.vnet.net>
  • Reply-to: Vladimir Bondarenko <vvb at mail.strace.net>
  • Sender: owner-wri-mathgroup at wolfram.com

flavionet at libero.it (Flavio Cimolin)wrote on Wednesday, November 13, 2002, 2:11:47 AM :

FC> Hi, I have a problem with the following improper integral:

FC> Integrate[x^a Log[x]/(x^2+1),{x,0,+Infinity}]

FC> which I know doesn't give 0 for {-1<a<1}.
FC> Why is the result wrong?

FC> For example:

FC> NIntegrate[x^.5 Log[x]/(x^2 + 1), {x, 0, 100000}]

FC> give 3.40397

FC> What's wrong with that?


You've come across a bug the long-liver ;-)

It is absent in the Version Windows 387 2.2 (April 9, 1993)
written by Alexei Bocharov, and Version Microsoft Windows 3.0
(April 25, 1997) written by Oleg Marichev and Victor Adamchik.

It looks like it was introduced in Version 4.0 for Microsoft
Windows (April 21, 1999), and has survived up to the current
Version 4.2 for Microsoft Windows (June 5, 2002).

Hopefully, thanks to you, it will be fixed in the next release.

Actually the symbolic answer should be

  If[-1 < Re[a] < 1, Pi^2 Sec[a Pi/2] Tan[a Pi/2]/4,
  Integrate[(x^a Log[x])/(1 + x^2), {x, 0, Infinity}]]

For example, let us set a = 3/4

  N[Pi^2*Sec[(a*Pi)/2]*Tan[(a*Pi)/2]/4 /. a -> 3/4, 78]
  NIntegrate[x^(3/4) Log[x]/(x^2 + 1), {x, 0, Infinity}, WorkingPrecision -> 500]

  15.5659552943718983805647319069251888094548594372508969754503565735702566414125
  15.5659552943718983805647319069251888094548594372508969754503565735702566414125

What about a = -9/10 ?

  N[Pi^2*Sec[(a*Pi)/2]*Tan[(a*Pi)/2]/4 /. a -> -9/10, 55]
  NIntegrate[x^(-9/10) Log[x]/(x^2 + 1), {x, 1/10^1000, Infinity}, WorkingPrecision -> 500]

  -99.58519965628388392337226875325913764330443917916599896
  -99.58519965628388392337226875325913764330443917916599896
  
I have added to CC the Technical support's address.


Best wishes,

Vladimir Bondarenko
Mathematical and Production Director
Symbolic Testing Group
Email:  vvb at mail.strace.net

Web  :  http://www.CAS-testing.org/ (under development, 95% ready)
        http://maple.bug-list.org/  (under development, 20% ready)

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