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Re: Chebyshev's Identity

• To: mathgroup at smc.vnet.net
• Subject: [mg46502] Re: [mg46494] Chebyshev's Identity
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 20 Feb 2004 22:58:35 -0500 (EST)
• References: <200402201153.GAA28293@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Ravinder Kumar B. wrote:
> Friends,
> 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.


First, it helps to use correct syntax and spell out whatever assumptions 
you may have in mind. Below I make some guesses that may or may not 
reflect your actual needs. I'll assume from the structure of the matrix 
that "iy" was meant to be "I*y". We start with the matrix:

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

Now raise it to the power n:

matpowern = MatrixPower[mat, n];

(Note that mat^n does something else entirely; Power is Listable so it 
simply takes each element to the nth power).

Again from the structure I will assume x and y are meant to be real 
valued. Also I'll assume you have in mind that n takes on integer values 
  though most likely this assumption makes no difference in Limit.

lp = Limit[matpowern, n->Infinity,
   Assumptions->{Element[{x,y},Reals],Element[n,Integers]}];

Here is what we have.

In[12]:= InputForm[lp]

Out[12]//InputForm=
{{(E^((-360*x^4*Sqrt[-x^2 - y^2] - 720*x^2*y^2*Sqrt[-x^2 - y^2] -
        360*y^4*Sqrt[-x^2 - y^2])/(360*(x^2 + y^2)^2))*
     (6*x^4 + 6*E^(2*Sqrt[-x^2 - y^2])*x^4 + 12*x^2*y^2 +
      12*E^(2*Sqrt[-x^2 - y^2])*x^2*y^2 + 6*y^4 + 6*E^(2*Sqrt[-x^2 - y^2])*
       y^4 - (6*I)*x^2*y*Sqrt[-x^2 - y^2] + (6*I)*E^(2*Sqrt[-x^2 - 
y^2])*x^2*y*
       Sqrt[-x^2 - y^2] - (6*I)*y^3*Sqrt[-x^2 - y^2] +
      (6*I)*E^(2*Sqrt[-x^2 - y^2])*y^3*Sqrt[-x^2 - y^2]))/(12*(x^2 + 
y^2)^2),
   ((-1 + E^(2*Sqrt[-x^2 - y^2]))*x)/(2*E^Sqrt[-x^2 - y^2]*Sqrt[-x^2 - 
y^2])},
  {-((-1 + E^(2*Sqrt[-x^2 - y^2]))*x)/(2*E^Sqrt[-x^2 - y^2]*Sqrt[-x^2 - 
y^2]),
   (E^((-360*x^4*Sqrt[-x^2 - y^2] - 720*x^2*y^2*Sqrt[-x^2 - y^2] -
        360*y^4*Sqrt[-x^2 - y^2])/(360*(x^2 + y^2)^2))*
     (6*x^4 + 6*E^(2*Sqrt[-x^2 - y^2])*x^4 + 12*x^2*y^2 +
      12*E^(2*Sqrt[-x^2 - y^2])*x^2*y^2 + 6*y^4 + 6*E^(2*Sqrt[-x^2 - y^2])*
       y^4 + (6*I)*x^2*y*Sqrt[-x^2 - y^2] - (6*I)*E^(2*Sqrt[-x^2 - 
y^2])*x^2*y*
       Sqrt[-x^2 - y^2] + (6*I)*y^3*Sqrt[-x^2 - y^2] -
      (6*I)*E^(2*Sqrt[-x^2 - y^2])*y^3*Sqrt[-x^2 - y^2]))/(12*(x^2 + 
y^2)^2)}}

A brief numerical check:

In[16]:= InputForm[(lp /. {x->2.1, y->-3.7}) - (matpowern /. {x->2.1, 
y->-3.7,n->1000000})]

Out[16]//InputForm=
{{6.763487900851572*^-7 + 7.423018731911313*^-7*I,
   -4.388215642303628*^-6 + 1.6381658794886671*^-6*I},
  {-4.213425174315333*^-7 + 1.6381227114990317*^-6*I,
   6.763889269234546*^-7 - 7.424337283845617*^-7*I}}

For your original question, you might find information regarding 
Chebyshev identities at the MathWorld web site.


Daniel Lichtblau
Wolfram Research

• References:
  • Chebyshev's Identity
    • From: "Ravinder Kumar B." <ravi@crest.ernet.in>