[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**Re: reference manager in Mathematica?**
Next by Date:
**Re: Help generalizing Liouville's Polynomial Identity**
Previous by thread:
**Re: Re: Incorrect symbolic improper integral**
Next by thread:
**Re: Re: Incorrect symbolic improper integral**
| |