Fungrim entry: ef4b53

\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:

n \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
`Abs`	$\left\|z\right\|$	Absolute value
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	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))))

Topics using this entry

Legendre polynomials