Re: Integrating Abs[Sin[]^2]
- To: mathgroup at smc.vnet.net
- Subject: [mg38850] Re: [mg38830] Integrating Abs[Sin[]^2]
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Thu, 16 Jan 2003 03:18:53 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
In this case avoiding Abs by first converting the integrand (using
ComplexExpand) to a different form gives the right answer:
ComplexExpand[Integrate[ComplexExpand[Abs[Sin[k*x]]^2, {k},
TargetFunctions -> {Conjugate}], {x, 0, 1}], {k},
TargetFunctions -> {Re, Im}]
-(Sin[2*Re[k]]/(4*Re[k])) + Sinh[2*Im[k]]/(4*Im[k])
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Wednesday, January 15, 2003, at 04:19 PM, Jos R Bergervoet wrote:
> A strange result appeared when using
>
> Mathematica 4.1 for Linux
> Copyright 1988-2000 Wolfram Research, Inc.
> -- Motif graphics initialized --
>
> 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?
> What should I do to circumvent such errors?
>
> Thanks in adv.,
>
> -- Jos <jos.bergervoet at philips.no_s_p_a_m.com>
>
> PS: For those interested, the correct analytical result is:
>
> (Sinh[2Im[k]]/Im[k] - Sin[2Re[k]]/Re[k]) / 4
>
>
>