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Fungrim entry: 2a4b9d

Un ⁣(z)(z+z2+1)n\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}
Assumptions:nZ0andzCn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("2a4b9d"),
    Formula(LessEqual(Abs(ChebyshevU(n, z)), Pow(Add(Abs(z), Sqrt(Add(Pow(Abs(z), 2), 1))), n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

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

2019-10-05 13:11:19.856591 UTC