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Chebyshev Polynomials

Posted 11 years ago
Friends,

I know that Chebyshev polynomials of degree n can put in the following form: Tn(x)=Cos[n*arccos[ x] ], x pertaining to  [-1,1]. When I create the command

Plot[{ChebyshevT[2, x], ChebyshevT[3, x], ChebyshevT[4, x]}, {x, -2, 2}, PlotLegend -> {"T2", "T3", "T4"}, LegendPosition -> {1, 0}]

It works well and I think it must not work well because "ArcCos for x>1" makes no sense. Anybody could help me?

Thank you,
Ana
POSTED BY: Ana Squadri
2 Replies
Posted 11 years ago
You are completely right. Really the trigonometric definition is just valid between -1 and 1 but exist the polynomial for other values. Thank you.
POSTED BY: Ana Squadri
Being polynomials, they are defined for any x. 
In[21]:= ChebyshevT[3, x]

Out[21]= -3 x + 4 x^3
Besides the usual recurrence definition they can also be defined by the above trigonometric formula, which indeed makes sense only for x between -1 and 1, but nevertheless determines the polynomial uniquely (because there is only one n-th degree polynomial passing through n+1 distinct points).
POSTED BY: Ilian Gachevski
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