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Fungrim entry: ef4b53

Pn ⁣(z)Pn ⁣(zi)(z+1+z2)n\left|P_{n}\!\left(z\right)\right| \le \left|P_{n}\!\left(\left|z\right| i\right)\right| \le {\left(\left|z\right| + \sqrt{1 + {\left|z\right|}^{2}}\right)}^{n}
Assumptions:nZ0andzCn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
\left|P_{n}\!\left(z\right)\right| \le \left|P_{n}\!\left(\left|z\right| i\right)\right| \le {\left(\left|z\right| + \sqrt{1 + {\left|z\right|}^{2}}\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
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
ConstIii Imaginary unit
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("ef4b53"),
    Formula(LessEqual(Abs(LegendrePolynomial(n, z)), Abs(LegendrePolynomial(n, Mul(Abs(z), ConstI))), Pow(Add(Abs(z), Sqrt(Add(1, Pow(Abs(z), 2)))), n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

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

2019-08-19 14:38:23.809000 UTC