Question #140930
(a) Using the formula Tn(x) = cos(n cos−1 x), n ≥ 0, find the Chebyshev
polynomials T0(x), T1(x), T2(x), T3(x), and T4(x).
1
Expert's answer
2020-11-03T14:57:24-0500

T0(x)=cos(0cos1x)=1T_0(x)=cos(0cos^{-1}x)=1

T1(x)=cos(cos1x)=xT_1(x)=cos(cos^{-1}x)=x

T2(x)=cos(2cos1x)=2cos2(cos1x)1=2x21T_2(x)=cos(2cos^{-1}x)=2cos^2(cos^{-1}x)-1=2x^2-1

T3(x)=cos(3cos1x)=4cos3(cos1x)3cos(cos1x)=4x33xT_3(x)=cos(3cos^{-1}x)=4cos^3 (cos^{-1}x)-3 cos (cos^{-1}x)= 4x^3-3x

T4(x)=cos(4cos1x)=2cos2(2cos1x)1=2(2cos2(cos1x)1)21=8cos4(cos1x)8cos2(cos1x)+21=8x48x2+1T_4(x)=cos(4cos^{-1}x)=2cos^2(2 cos^{-1}x) -1=2(2cos^2(cos^{-1}x)-1)^2-1=8cos^4 (cos^{-1}x)-8cos^2 (cos^{-1}x)+2-1= 8x^4-8x^2+1


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