Re: Re: Simplification of definite integral?
- To: mathgroup at smc.vnet.net
- Subject: [mg40765] Re: [mg40720] Re: Simplification of definite integral?
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 16 Apr 2003 01:38:45 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Here is another way to get the right answer for the case when d is real
and without using the PrincipalValue option:
FullSimplify[Integrate[ComplexExpand[
TrigToExp[(Sin[x - d]/(x - d))*(Sin[x + d]/(x + d))]],
{x, -Infinity, Infinity}]]
(Pi*Cos[d]*Sin[d])/d
In the non-real case the following:
FullSimplify[Integrate[TrigReduce[(Sin[x - d]/(x - d))*
(Sin[x + d]/(x + d))], {x, -Infinity, Infinity},
Assumptions -> Arg[d^2] != 0]]
(Sqrt[-(1/d^2)]*Sqrt[-d]*Pi*Cos[d]*Sin[d])/Sqrt[d]
also seems to be correct while the same thing without TrigReduce
appears wrong.
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Tuesday, April 15, 2003, at 04:56 pm, Dr. Wolfgang Hintze wrote:
> 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
>>>
>>>
>>>
>>>
>>>
>>
>>
>
>
>
>