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MathGroup Archive 2004

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Re: Chebyshev's Identity

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46515] Re: Chebyshev's Identity
  • From: bobhanlon at aol.com (Bob Hanlon)
  • Date: Fri, 20 Feb 2004 22:58:58 -0500 (EST)
  • References: <c14spn$rna$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

By iy I assume that you meant I*y

M=({{Cos[x/n]*Exp[-I*y/n],Sin[x/n]*Exp[-I*y/n]},
        {-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[I*y/n]}})^n

{{(Cos[x/n]/E^((I*y)/n))^n, (Sin[x/n]/E^((I*y)/n))^n}, 
  {((-E^(iy/n))*Sin[x/n])^n, (E^((I*y)/n)*Cos[x/n])^n}}

Limit[M, n->Infinity]

{{E^((-I)*y), 0}, {0, E^(I*y)}}

Note that each term in the original matrix was individually raised to power n.
By "matrix raised to power n" you may have meant

M=MatrixPower[{{Cos[x/n]*Exp[-I*y/n],
          Sin[x/n]*Exp[-I*y/n]},{-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[I*y/n]}},n];

Limit[M, n->Infinity]//FullSimplify

{{Cosh[Sqrt[-x^2 - y^2]] - (I*y*Sinh[Sqrt[-x^2 - y^2]])/
     Sqrt[-x^2 - y^2], (x*Sinh[Sqrt[-x^2 - y^2]])/
    Sqrt[-x^2 - y^2]}, 
  {-((x*Sinh[Sqrt[-x^2 - y^2]])/Sqrt[-x^2 - y^2]), 
   Cosh[Sqrt[-x^2 - y^2]] + (I*y*Sinh[Sqrt[-x^2 - y^2]])/
     Sqrt[-x^2 - y^2]}}


Bob Hanlon

In article <c14spn$rna$1 at smc.vnet.net>, "Ravinder Kumar B."
<ravi at crest.ernet.in> wrote:

<< I have a (2x2) matrix raised to power n.

M = ({{Cos[x/n]*Exp[-iy/n],
Sin[x/n]*Exp[-iy/n]},{-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[iy/n]}})^n;

All I know at present is that this expression can be further
simplified analytically using Chebyshev's identity to much a simpler
expression in the limit n -> infinity.
I am unable to find any information regarding Chebyseb's identity and its
usage.
Could some one please tell me more about this identity and its usage in
solving above expression. Mathematica fails to do it analytically.


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