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MathGroup Archive 2005

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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)         
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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