Re: Simplification of definite integral?
- To: mathgroup at smc.vnet.net
- Subject: [mg40720] Re: Simplification of definite integral?
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Tue, 15 Apr 2003 03:56:58 -0400 (EDT)
- References: <b7dq8s$6a2$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Andrzej,
thanks for your hint. The final answer is what I expected from
mathematica (and know to be correct).
Best regards,
Wolfgang
Andrzej Kozlowski wrote:
> Mathematica has difficulties dealing with the (apparent) singularities
> at x==d and x == -d so if you try straight forward Integrate it want's
> you to assume that d non-real. However, you can get an answer probably
> closer to what you desire by setting the PrincipalValue option to True:
>
>
> Integrate[Sin[x-d]/(
> x-d) Sin[x+d]/(x+
> d),{x,-Infinity,Infinity},PrincipalValue->True,Assumptions->{d>0}]
>
>
> (Pi*Cos[d]*Sin[d])/d
>
> For example for d =1 we get:
>
>
> %/.d->1.
>
>
> 1.42832
>
> This is probably right, particularly that
>
>
> NIntegrate[(Sin[x - 1]/(x - 1))*(Sin[x + 1]/(x + 1)),
> {x, -Infinity, 1, Infinity}]
>
>
> NIntegrate::slwcon:Numerical integration converging too slowly; suspect
> one \
> of the following: singularity, value of the integration being 0,
> oscillatory \
> integrand, or insufficient WorkingPrecision. If your integrand is
> oscillatory \
> try using the option Method->Oscillatory in NIntegrate.
>
>
> NIntegrate::ncvb:NIntegrate failed to converge to prescribed accuracy
> after 7 \
> recursive bisections in x near x = 187.1757811919331`.
>
>
> 1.4283406894658994
>
>
> Andrzej Kozlowski
> Yokohama, Japan
> http://www.mimuw.edu.pl/~akoz/
> http://platon.c.u-tokyo.ac.jp/andrzej/
>
>
>
>
>
> On Sunday, April 13, 2003, at 03:17 pm, Dr. Wolfgang Hintze wrote:
>
>
>>How do I get a satisfactory result from mathematica for this function
>>
>>f[d]:=Integrate[Sin[x-d]/(x-d) Sin[x+d]/(x+d),{x,-Infinity,Infinity}]
>>
>>I tried
>>
>>f[d]//ComplexExpand
>>
>>and several assumptions but I didn't succeed. Any hints?
>>
>>Wolfgang
>>
>>
>>
>>
>>
>
>