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Re: Re: Simplification of definite integral?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg40727] Re: [mg40698] Re: Simplification of definite integral?
  • From: Dr Bob <majort at cox-internet.com>
  • Date: Tue, 15 Apr 2003 03:58:50 -0400 (EDT)
  • References: <b7avdo$qlg$1@smc.vnet.net> <200304140801.EAA06347@smc.vnet.net>
  • Reply-to: majort at cox-internet.com
  • Sender: owner-wri-mathgroup at wolfram.com

If you remove GenerateConditions->False, we can get this:

Integrate[Sin[x - d]/(x - d) Sin[x + d]/(x + d), {x, -Infinity, Infinity}];
FullSimplify@%[[2, 2]]

(Cos[2*d]*(-Log[-(1/d)] + Log[1/d]))/(2*d)

just as you did.  But it's inappropriate to simplify this by assuming that 
d is real, since the conditions conveniently omitted that way (or using 
GenerateConditions->False) are

Arg[d] â?  0 && Im[d] â?  0

and so, d can't be real.  And yet, your answer is correct:

Integrate[Sin[x - d]/(x - d) Sin[x + d]/(
        x + d), {x, -Infinity, Infinity}, PrincipalValue -> True];
Simplify[%, d > 0]

(Pi*Cos[d]*Sin[d])/d

Bobby

On Mon, 14 Apr 2003 04:01:22 -0400 (EDT), Bob Hanlon <bobhanlon at aol.com> 
wrote:

> f[d_] := Evaluate[FullSimplify[
> Integrate[Sin[x-d]/(x-d) Sin[x+d]/(x+d),{x,-Infinity,Infinity}, 
> GenerateConditions->False]]]
>
> f[d]
>
> (Cos[2*d]*(Log[-(1/d)] - Log[1/d]))/(2*d)
>
> f[d]//PowerExpand
>
> (I*Pi*Cos[2*d])/(2*d)
>
> Simplify[f[d], Element[d, Reals]&&d>0]
>
> (I*Pi*Cos[2*d])/(2*d)
>
>
> Bob Hanlon
>
> In article <b7avdo$qlg$1 at smc.vnet.net>, "Dr. Wolfgang Hintze" 
> <weh at snafu.de>
> wrote:
>
> << Subject:	Simplification of definite integral?
> From:		"Dr. Wolfgang Hintze" <weh at snafu.de>
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> Date:		Sun, 13 Apr 2003 06:17:28 +0000 (UTC)
>
> 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 >><BR><BR>
>
>



-- 
majort at cox-internet.com
Bobby R. Treat



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