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 >> >> >> >> >> > >