Re: Bounds for Trig expression

*To*: mathgroup at smc.vnet.net*Subject*: [mg54683] Re: Bounds for Trig expression*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Sun, 27 Feb 2005 01:29:07 -0500 (EST)*Organization*: The University of Western Australia*References*: <cvk3hc$d6p$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cvk3hc$d6p$1 at smc.vnet.net>, "Hugh" <h.g.d.goyder at cranfield.ac.uk> wrote: > I wish to find bounds for the complex trig expression ee below. The > expression depends on the real variables Sr, a and k which lie in the > range > 0 < Sr <1 > 0 < a < 1 > 0 < k > A blind numerical evaluation of many values, plotted below, suggests > that the real part is bounded by (-32, 16) while the imaginary part is > bounded by approximately (-26, 26). I am happy bounding in a > rectangular region on the complex plane although the numerical plot > suggests an elliptical region. > > Is there a non numerical approach to finding the bounds? Possibly by > replacing Cos and Sin by all permutation of + and - 1? A semi-numerical approach is to use Interval. > I have more expressions like this to tackle so I would like an approach > that can be generalized. > > ee = (-1 + Cos[a*k] + I*Sin[a*k])*(6 - 3*Sr + 3*(2 + Sr)*Cos[(-2 + > a)*k] - (2 + Sr)*Cos[2*(-1 + a)*k] - > 2*Cos[a*k] + Sr*Cos[a*k] - 6*I*Sin[(-2 + a)*k] - 3*I*Sr*Sin[(-2 + > a)*k] + 2*I*Sin[2*(-1 + a)*k] + > I*Sr*Sin[2*(-1 + a)*k] - 2*I*Sin[a*k] + I*Sr*Sin[a*k]); Entering FullSimplify[ee /. {a -> Interval[{0, 1}], Sr -> Interval[{0, 1}], k -> Interval[{0, 2 Pi}]}] gives a cruder bound. This bound is not tight because the intervals are treated as independent. Also, consider expressions like FullSimplify[Exp[I Interval[{0, 2 Pi}]]] The resulting rectangular bound encloses the unit circle. Finally, I note that ee can be written as (z^a - 1) ((Sr + 2) (3 - z^(-a)) z^(2 - a) + (Sr - 2) (z^a - 3)) where z = Exp[I k]. Perhaps this form can be used to find a tighter bound? Cheers, Paul -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul

**Follow-Ups**:**Re: Re: Bounds for Trig expression***From:*Chris Chiasson <chris.chiasson@gmail.com>