Re: Integrating Abs[Sin[]^2]
- To: mathgroup at smc.vnet.net
- Subject: [mg38860] Re: Integrating Abs[Sin[]^2]
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 16 Jan 2003 03:20:40 -0500 (EST)
- References: <b032m9$mv4$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Jos R Bergervoet <antispam at nospam.com> wrote:
> in computing the following:
>
> result = Integrate[ Abs[Sin[k x]]^2, {x,0,1}]
> N[ result /. k->I+1 ]
>
> (* Analytical approach gives 0.261044 + 0.616283 I, WRONG !!! *)
>
> k=I+1; NIntegrate[ Abs[Sin[k x]^2], {x,0,1}]
>
> (* Numerical check gives 0.679391 *)
>
> So why is the analytical result for |Sin[k x]|^2 wrong?
It might be of interest that another CAS which I use also gives both of
those results.
> What should I do to circumvent such errors?
One thing that works in Mathematica (as well as in the other CAS) is to
Integrate[ Abs[Sin[(a+b*I) x]]^2, {x,0,1}].
This gives (a*Sinh[2*b] - b*Sin[2*a]) / (4*a*b),
which agrees with your result below.
BTW, I'm mildly surprised that telling Mathematica explicitly that k should
be assumed to be complex in the integration doesn't work.
David
> PS: For those interested, the correct analytical result is:
>
> (Sinh[2Im[k]]/Im[k] - Sin[2Re[k]]/Re[k]) / 4