Re: Bounds for Trig expression

• To: mathgroup at smc.vnet.net
• Subject: [mg54674] Re: [mg54603] Bounds for Trig expression
• From: Chris Chiasson <chris.chiasson at gmail.com>
• Date: Fri, 25 Feb 2005 01:19:51 -0500 (EST)
• References: <200502240821.DAA13179@smc.vnet.net>
• Reply-to: Chris Chiasson <chris.chiasson at gmail.com>
• Sender: owner-wri-mathgroup at wolfram.com

```Does this work for you?

Sr = Interval[{0, 1}]
a = Interval[{0, 1}]
k = Interval[{0, â??}]
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]) // Simplify

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