       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

```

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