Chebyshev Polynomial Calculator
Calculate Chebyshev polynomials T_n(x) and U_n(x) with roots, extrema, and trigonometric verification.
Input
Polynomial Form
T_5(x) = 16x⁵ - 20x³ + 5x
T_5(0.5)
0.5000000000
Trigonometric Verification
cos(5 × arccos(0.5)) = 0.5000000000
Sequence at x = 0.5
T_0=1.0000
T_1=0.5000
T_2=-0.5000
T_3=-1.0000
T_4=-0.5000
T_5=0.5000
Roots of T_5(x) (Chebyshev Nodes)
0.9510570.5877850.000000-0.587785-0.951057
Recurrence Relation
T_n(x) = 2x·T_{n-1}(x) - T_{n-2}(x)
with T_0=1, T_1=x
Orthogonality
Weight function w(x) = 1/√(1-x²) on [-1, 1]
Chebyshev Polynomials
First Kind T_n
- T_n(cos θ) = cos(nθ)
- |T_n(x)| ≤ 1 for |x| ≤ 1
- Minimal deviation from zero
Applications
- Polynomial interpolation
- Numerical integration (Gauss-Chebyshev)
- Filter design