Fungrim home page

Fungrim entry: 35e13b

Un(x)=(n+1)Tn+1 ⁣(x)xUn ⁣(x)x21U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\right)}{{x}^{2} - 1}
Assumptions:nZ  and  xC{1,1}n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}
U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\right)}{{x}^{2} - 1}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x)), Div(Sub(Mul(Add(n, 1), ChebyshevT(Add(n, 1), x)), Mul(x, ChebyshevU(n, x))), Sub(Pow(x, 2), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(-1, 1))))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC