Bounds for Trig expression

*To*: mathgroup at smc.vnet.net*Subject*: [mg54603] Bounds for Trig expression*From*: "Hugh" <h.g.d.goyder at cranfield.ac.uk>*Date*: Thu, 24 Feb 2005 03:21:07 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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

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

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