Mathematical Experiment 6

exp6.mws
  1.  
    1. Use the command ``expand'' to express cos(kx) as a polynomial function of cosx for k = 1,2,3,4,5,6.
    2. The Chebyshev polynomials (of the first kind) are defined recursively by
      3. T0(x) º 1, T1(x) = x and Tn+1(x) = 2xTn(x)-Tn-1(x) forn³ 1.

      Construct Tn(x) from this definition.

    3. Show that Tn(cos(x)) = cosnx.
    4. Chebyshev polynomials form a series of orthogonal polynomials. Compare your result with the function T(n,x) given in the package ``orthopoly''.
    5. Draw the graphs of T1, T2, T3,...,T6 on the square [-1,1] × [-1,1] .
    6. Apply the command chebyshev to find the Chebyshev expansion of the polynomials 2n-1xn,n = 0,1,¼,6.
    7. Use the command ``combine'' to express coskx as a linear combination of 2n-1cos(nx),k = 1,2,3,4,5,6.
  2.  
    1. Use the command ``expand'' to express sin(kx) in the form Uk(cosx)sinx for some polynomial function Uk, for k = 1,2,3,4,5,6.
    2. The Chebyshev polynomials of the second kind are defined recursively by
      3. U0(x) º 1 , U1(x) = 2x  and Un+1(x) = 2xUn(x) -Un-1(x) forn³ 1. 
    3. Construct Un(x) from this definition.
    4. Show that Un(sin(x)) = sin nx.
    5. Chebyshev polynomials Un form a series of orthogonal polynomials. Compare your result with the function U(n,x) given in the package ``orthopoly''.
    6. Draw the graphs of U1, U2, U3,...,U6.
    7. Use the command ``combine'' to express sinkx as a linear combination of 2n-1sin(nx),k = 1,3,5,7,9.
  3. Let A be a 2×2 matrix. Suppose that A has eigenvalues a and b. It is known that the determinant of A is m = ab and the trace of A is t = a+b. Since an and bn are the eigenvalues of An, it follows that the determinant of An is anbn while the trace of An is Tn = an+bn.
    1. Find the recurrence relations satisfied by Tn for n ³ 0.
    2. Express Tn as a function of t and m for n = 0,1,2,...,20.
  4. Find the pattern of
     
    for n = 1,2,3,···. Use the spreadsheet to find its expansion.
  1. The shifted Chebyshev polynomials are given by
    2. Tk*(x) = Tk(2x-1) 
    1. Find the recursive relation satisfied by Tk*.
    2. Compute the coefficients of Tk* for k = 1, 2, ···, 10.
    3. Express 22n-1xn as a linear combination of Tk* for n = 1,2,···,12.
  2. Let the polynomials Cn(x) be given recursively by
    3. C0(x) = 2 , C1(x) = x , Cn+1(x) = xCn(x)-Cn-1(x) for n ³ 1. 
    1. List the coefficients of Cn(x) for n = 0,1,2,···,12.
    2. Express xn as a linear combination of Ck(x) for n = 0,1,2,···,12.
  3. Let the polynomials Sn(x) be given recursively by
    4. S0(x) = 1 , S1(x) = x , Sn+1(x) = xSn(x)-Sn-1(x) for n ³ 1. 
    1. List the coefficients of Sn(x) for n = 0,1,2,···,12.
    2. Express xn as a linear combination of Sk(x) for n = 0,1,2,···,12.
exp6.tex