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>
- Chebyshev's Identity