Re: Bounds for Trig expression

*To*: mathgroup at smc.vnet.net*Subject*: [mg54688] Re: [mg54603] Bounds for Trig expression*From*: Chris Chiasson <chris.chiasson at gmail.com>*Date*: Sun, 27 Feb 2005 01:29:12 -0500 (EST)*References*: <200502240821.DAA13179@smc.vnet.net>*Reply-to*: Chris Chiasson <chris.chiasson at gmail.com>*Sender*: owner-wri-mathgroup at wolfram.com

whoops -- k is on the interval from zero to infinity ... not from zero to strange characters :-) On Thu, 24 Feb 2005 03:21:07 -0500 (EST), 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? > > 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]); > > d1 = Partition[Flatten[Table[({Re[#1], Im[#1]} & )[ee], {k, 0, 100*Pi, > 0.3*Pi}, {a, 0, 1, 0.1}, > {Sr, 0, 1, 0.1}]], 2]; > > ListPlot[d1, AspectRatio -> Automatic]; > > ({Max[#1], Min[#1]} & )[d1[[All,1]]] > > ({Max[#1], Min[#1]} & )[d1[[All,2]]] > > Thanks > > Hugh Goyder > > -- Chris Chiasson Kettering University Mechanical Engineering Graduate Student 1 810 265 3161

**References**:**Bounds for Trig expression***From:*"Hugh" <h.g.d.goyder@cranfield.ac.uk>