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) Voice: (380)-652-447325 Mon-Fri 6 a.m. - 3 p.m. GMT ICQ : 173050619 Mail : 76 Zalesskaya Str, Simferopol, Crimea, Ukraine
- References:
- Improper integral
- From: flavionet@libero.it (Flavio Cimolin)
- Improper integral