Homework answers / question archive / An Nth degree Chebyshev polynomial is defined by (C_N)(x) = cos(Ncos^-1 (x)) = cosh(Ncosh^-1 (x)) i) Show that C_N(x) is real for real values of x ii) Show that the two expressions are equal (hint: let w = cos^-1(x) + work with cos expression (middle term)) iii) Show that C_N (x) satisfies Co(x) = 1, c(x) = x and C_N (x) = 2x(C_N-1) (x) - C_(N-2) (x), for N = 2, 3

An Nth degree Chebyshev polynomial is defined by (C_N)(x) = cos(Ncos^-1 (x)) = cosh(Ncosh^-1 (x)) i) Show that C_N(x) is real for real values of x ii) Show that the two expressions are equal (hint: let w = cos^-1(x) + work with cos expression (middle term)) iii) Show that C_N (x) satisfies Co(x) = 1, c(x) = x and C_N (x) = 2x(C_N-1) (x) - C_(N-2) (x), for N = 2, 3

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An Nth degree Chebyshev polynomial is defined by (C_N)(x) = cos(Ncos^-1 (x)) = cosh(Ncosh^-1 (x))

i) Show that C_N(x) is real for real values of x
ii) Show that the two expressions are equal (hint: let w = cos^-1(x) + work with cos expression (middle term))
iii) Show that C_N (x) satisfies Co(x) = 1, c(x) = x and C_N (x) = 2x(C_N-1) (x) - C_(N-2) (x), for N = 2, 3 ...