Legendre polynomials

9bdf22

P_{0}\!\left(z\right) = 1

Assumptions:

z \in \mathbb{C}

TeX:

P_{0}\!\left(z\right) = 1

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9bdf22"),
    Formula(Equal(LegendrePolynomial(0, z), 1)),
    Variables(z),
    Assumptions(Element(z, CC)))

217521

P_{1}\!\left(z\right) = z

Assumptions:

z \in \mathbb{C}

TeX:

P_{1}\!\left(z\right) = z

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("217521"),
    Formula(Equal(LegendrePolynomial(1, z), z)),
    Variables(z),
    Assumptions(Element(z, CC)))

d77f0a

P_{2}\!\left(z\right) = \frac{1}{2} \left(3 {z}^{2} - 1\right)

Assumptions:

z \in \mathbb{C}

TeX:

P_{2}\!\left(z\right) = \frac{1}{2} \left(3 {z}^{2} - 1\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d77f0a"),
    Formula(Equal(LegendrePolynomial(2, z), Mul(Div(1, 2), Sub(Mul(3, Pow(z, 2)), 1)))),
    Variables(z),
    Assumptions(Element(z, CC)))

9b7f05

P_{3}\!\left(z\right) = \frac{1}{2} \left(5 {z}^{3} - 3 z\right)

Assumptions:

z \in \mathbb{C}

TeX:

P_{3}\!\left(z\right) = \frac{1}{2} \left(5 {z}^{3} - 3 z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9b7f05"),
    Formula(Equal(LegendrePolynomial(3, z), Mul(Div(1, 2), Sub(Mul(5, Pow(z, 3)), Mul(3, z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

a17386

P_{4}\!\left(z\right) = \frac{1}{8} \left(35 {z}^{4} - 30 {z}^{2} + 3\right)

Assumptions:

z \in \mathbb{C}

TeX:

P_{4}\!\left(z\right) = \frac{1}{8} \left(35 {z}^{4} - 30 {z}^{2} + 3\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a17386"),
    Formula(Equal(LegendrePolynomial(4, z), Mul(Div(1, 8), Add(Sub(Mul(35, Pow(z, 4)), Mul(30, Pow(z, 2))), 3)))),
    Variables(z),
    Assumptions(Element(z, CC)))

13f971

P_{5}\!\left(z\right) = \frac{1}{8} \left(63 {z}^{5} - 70 {z}^{3} + 15 z\right)

Assumptions:

z \in \mathbb{C}

TeX:

P_{5}\!\left(z\right) = \frac{1}{8} \left(63 {z}^{5} - 70 {z}^{3} + 15 z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("13f971"),
    Formula(Equal(LegendrePolynomial(5, z), Mul(Div(1, 8), Add(Sub(Mul(63, Pow(z, 5)), Mul(70, Pow(z, 3))), Mul(15, z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

a7ac51

P_{n}\!\left(1\right) = 1

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

P_{n}\!\left(1\right) = 1

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a7ac51"),
    Formula(Equal(LegendrePolynomial(n, 1), 1)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

3df748

P_{n}\!\left(-1\right) = {\left(-1\right)}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

P_{n}\!\left(-1\right) = {\left(-1\right)}^{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3df748"),
    Formula(Equal(LegendrePolynomial(n, -1), Pow(-1, n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

674afa

P_{2 n}\!\left(0\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

P_{2 n}\!\left(0\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("674afa"),
    Formula(Equal(LegendrePolynomial(Mul(2, n), 0), Mul(Div(Pow(-1, n), Pow(4, n)), Binomial(Mul(2, n), n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

85eebc

P_{2 n + 1}\!\left(0\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

P_{2 n + 1}\!\left(0\right) = 0

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("85eebc"),
    Formula(Equal(LegendrePolynomial(Add(Mul(2, n), 1), 0), 0)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

0010f3

P_{n}\!\left(-z\right) = {\left(-1\right)}^{n} P_{n}\!\left(z\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(-z\right) = {\left(-1\right)}^{n} P_{n}\!\left(z\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0010f3"),
    Formula(Equal(LegendrePolynomial(n, Neg(z)), Mul(Pow(-1, n), LegendrePolynomial(n, z)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

367ac2

\left(n + 1\right) P_{n + 1}\!\left(z\right) - \left(2 n + 1\right) z P_{n}\!\left(z\right) + n P_{n - 1}\!\left(z\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left(n + 1\right) P_{n + 1}\!\left(z\right) - \left(2 n + 1\right) z P_{n}\!\left(z\right) + n P_{n - 1}\!\left(z\right) = 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("367ac2"),
    Formula(Equal(Add(Sub(Mul(Add(n, 1), LegendrePolynomial(Add(n, 1), z)), Mul(Mul(Add(Mul(2, n), 1), z), LegendrePolynomial(n, z))), Mul(n, LegendrePolynomial(Sub(n, 1), z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC))))

27688e

\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("27688e"),
    Formula(Equal(Add(Sub(Mul(Sub(1, Pow(z, 2)), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 2))), Mul(Mul(2, z), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 1)))), Mul(Mul(n, Add(n, 1)), LegendrePolynomial(n, z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

925fdf

\left(1 - {z}^{2}\right) P'_{n}(z) + n z P_{n}\!\left(z\right) - n P_{n - 1}\!\left(z\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left(1 - {z}^{2}\right) P'_{n}(z) + n z P_{n}\!\left(z\right) - n P_{n - 1}\!\left(z\right) = 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("925fdf"),
    Formula(Equal(Sub(Add(Mul(Sub(1, Pow(z, 2)), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 1))), Mul(Mul(n, z), LegendrePolynomial(n, z))), Mul(n, LegendrePolynomial(Sub(n, 1), z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC))))

d84519

\sum_{n=0}^{\infty} P_{n}\!\left(x\right) {z}^{n} = \frac{1}{\sqrt{1 - 2 x z + {z}^{2}}}

Assumptions:

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=0}^{\infty} P_{n}\!\left(x\right) {z}^{n} = \frac{1}{\sqrt{1 - 2 x z + {z}^{2}}}

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("d84519"),
    Formula(Equal(Sum(Mul(LegendrePolynomial(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(1, Sqrt(Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2)))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

4cfeac

P_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

Alternative assumptions:

z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}

n \in \mathbb{Z}_{\ge 0}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4cfeac"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Mul(Pow(2, n), Factorial(n))), ComplexDerivative(Pow(Sub(Pow(t, 2), 1), n), For(t, z, n))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0))), Element(z, CC)))

e36542

\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

TeX:

\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`KroneckerDelta`	$\delta_{(x,y)}$	Kronecker delta
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e36542"),
    Formula(Equal(Integral(Mul(LegendrePolynomial(n, x), LegendrePolynomial(m, x)), For(x, -1, 1)), Mul(Div(2, Add(Mul(2, n), 1)), KroneckerDelta(n, m)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

c5dd9b

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{n} {{n \choose k}}^{2} {\left(z - 1\right)}^{n - k} {\left(z + 1\right)}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{n} {{n \choose k}}^{2} {\left(z - 1\right)}^{n - k} {\left(z + 1\right)}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c5dd9b"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Pow(2, n)), Sum(Mul(Mul(Pow(Binomial(n, k), 2), Pow(Sub(z, 1), Sub(n, k))), Pow(Add(z, 1), k)), For(k, 0, n))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

f0569a

P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f0569a"),
    Formula(Equal(LegendrePolynomial(n, z), Sum(Mul(Mul(Binomial(n, k), Binomial(Add(n, k), k)), Pow(Div(Sub(z, 1), 2), k)), For(k, 0, n)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

7a85b7

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7a85b7"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Pow(2, n)), Sum(Mul(Mul(Mul(Pow(-1, k), Binomial(n, k)), Binomial(Sub(Mul(2, n), Mul(2, k)), n)), Pow(z, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

9395fc

P_{n}\!\left(z\right) = \,{}_2F_1\!\left(-n, n + 1, 1, \frac{1 - z}{2}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(z\right) = \,{}_2F_1\!\left(-n, n + 1, 1, \frac{1 - z}{2}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9395fc"),
    Formula(Equal(LegendrePolynomial(n, z), Hypergeometric2F1(Neg(n), Add(n, 1), 1, Div(Sub(1, z), 2)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

f55f0a

P_{n}\!\left(z\right) = {2 n \choose n} {\left(\frac{z}{2}\right)}^{n} \,{}_2F_1\!\left(-\frac{n}{2}, \frac{1 - n}{2}, \frac{1}{2} - n, \frac{1}{{z}^{2}}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

P_{n}\!\left(z\right) = {2 n \choose n} {\left(\frac{z}{2}\right)}^{n} \,{}_2F_1\!\left(-\frac{n}{2}, \frac{1 - n}{2}, \frac{1}{2} - n, \frac{1}{{z}^{2}}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f55f0a"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Mul(Binomial(Mul(2, n), n), Pow(Div(z, 2), n)), Hypergeometric2F1(Neg(Div(n, 2)), Div(Sub(1, n), 2), Sub(Div(1, 2), n), Div(1, Pow(z, 2)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(0))))))

3c87b9

P_{n}\!\left(z\right) = {\left(\frac{z - 1}{2}\right)}^{n} \,{}_2F_1\!\left(-n, -n, 1, \frac{z + 1}{z - 1}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

TeX:

P_{n}\!\left(z\right) = {\left(\frac{z - 1}{2}\right)}^{n} \,{}_2F_1\!\left(-n, -n, 1, \frac{z + 1}{z - 1}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("3c87b9"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Pow(Div(Sub(z, 1), 2), n), Hypergeometric2F1(Neg(n), Neg(n), 1, Div(Add(z, 1), Sub(z, 1)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))

6cd4a1

P_{2 n}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n} \,{}_2F_1\!\left(-n, n + \frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{2 n}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n} \,{}_2F_1\!\left(-n, n + \frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("6cd4a1"),
    Formula(Equal(LegendrePolynomial(Mul(2, n), z), Mul(Mul(Div(Pow(-1, n), Pow(4, n)), Binomial(Mul(2, n), n)), Hypergeometric2F1(Neg(n), Add(n, Div(1, 2)), Div(1, 2), Pow(z, 2))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

859445

P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {z}^{2}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {z}^{2}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("859445"),
    Formula(Equal(LegendrePolynomial(Add(Mul(2, n), 1), z), Mul(Mul(Mul(Mul(Div(Pow(-1, n), Pow(4, n)), Add(Mul(2, n), 1)), Binomial(Mul(2, n), n)), z), Hypergeometric2F1(Neg(n), Add(n, Div(3, 2)), Div(3, 2), Pow(z, 2))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

1ba9a5

\left|P_{n}\!\left(x\right)\right| \le 1

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

TeX:

\left|P_{n}\!\left(x\right)\right| \le 1

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1ba9a5"),
    Formula(LessEqual(Abs(LegendrePolynomial(n, x)), 1)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), LessEqual(-1, x, 1))))

155343

\left|P_{n}\!\left(x\right)\right| \le 2 I_{0}\!\left(2 n \sqrt{\frac{\left|x - 1\right|}{2}}\right) \le 2 {e}^{2 n \sqrt{\left|x - 1\right| / 2}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

TeX:

\left|P_{n}\!\left(x\right)\right| \le 2 I_{0}\!\left(2 n \sqrt{\frac{\left|x - 1\right|}{2}}\right) \le 2 {e}^{2 n \sqrt{\left|x - 1\right| / 2}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Exp`	${e}^{z}$	Exponential function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("155343"),
    Formula(LessEqual(Abs(LegendrePolynomial(n, x)), Mul(2, BesselI(0, Mul(Mul(2, n), Sqrt(Div(Abs(Sub(x, 1)), 2))))), Mul(2, Exp(Mul(Mul(2, n), Sqrt(Div(Abs(Sub(x, 1)), 2))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, RR))))

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))))

b786ad

\left|P'_{n}(x)\right| \le \frac{n \left(n + 1\right)}{2}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

TeX:

\left|P'_{n}(x)\right| \le \frac{n \left(n + 1\right)}{2}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b786ad"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 1))), Div(Mul(n, Add(n, 1)), 2))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), LessEqual(-1, x, 1))))

60ac50

\left|P'_{n}(x)\right| \le \frac{{2}^{3 / 2}}{\sqrt{\pi}} \frac{{n}^{1 / 2}}{{\left(1 - {x}^{2}\right)}^{3 / 4}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

TeX:

\left|P'_{n}(x)\right| \le \frac{{2}^{3 / 2}}{\sqrt{\pi}} \frac{{n}^{1 / 2}}{{\left(1 - {x}^{2}\right)}^{3 / 4}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("60ac50"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 1))), Mul(Div(Pow(2, Div(3, 2)), Sqrt(Pi)), Div(Pow(n, Div(1, 2)), Pow(Sub(1, Pow(x, 2)), Div(3, 4)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Less(-1, x, 1))))

59e5df

\left|P''_{n}(x)\right| \le \frac{\left(n - 1\right) n \left(n + 1\right) \left(n + 2\right)}{8}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

TeX:

\left|P''_{n}(x)\right| \le \frac{\left(n - 1\right) n \left(n + 1\right) \left(n + 2\right)}{8}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 \le x \le 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("59e5df"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 2))), Div(Mul(Mul(Mul(Sub(n, 1), n), Add(n, 1)), Add(n, 2)), 8))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), LessEqual(-1, x, 1))))

3b175b

\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

TeX:

\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3b175b"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 2))), Mul(Div(Pow(2, Div(5, 2)), Sqrt(Pi)), Div(Pow(n, Div(3, 2)), Pow(Sub(1, Pow(x, 2)), Div(5, 4)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Less(-1, x, 1))))

6476bd

\left|{P}^{(r)}_{n}(x)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

TeX:

\left|{P}^{(r)}_{n}(x)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6476bd"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, r))), Mul(Div(Pow(2, Add(r, Div(1, 2))), Sqrt(Pi)), Div(Pow(n, Sub(r, Div(1, 2))), Pow(Sub(1, Pow(x, 2)), Div(Add(Mul(2, n), 1), 4)))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(r, ZZGreaterEqual(0)), Less(-1, x, 1))))

40fa59

P_{n}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

P_{n}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("40fa59"),
    Formula(IsHolomorphic(LegendrePolynomial(n, z), ForElement(z, CC))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

d36fd7

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} P_{n}\!\left(z\right) = \left\{{\tilde \infty}\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} P_{n}\!\left(z\right) = \left\{{\tilde \infty}\right\}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d36fd7"),
    Formula(Equal(Poles(LegendrePolynomial(n, z), ForElement(z, Union(CC, Set(UnsignedInfinity)))), Set(UnsignedInfinity))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

99e62f

\operatorname{EssentialSingularities}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\operatorname{EssentialSingularities}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("99e62f"),
    Formula(Equal(EssentialSingularities(LegendrePolynomial(n, z), z, Union(CC, Set(UnsignedInfinity))), Set())),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

7680d3

\operatorname{BranchPoints}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\operatorname{BranchPoints}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7680d3"),
    Formula(Equal(BranchPoints(LegendrePolynomial(n, z), z, Union(CC, Set(UnsignedInfinity))), Set())),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

22a42f

\operatorname{BranchCuts}\!\left(P_{n}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\operatorname{BranchCuts}\!\left(P_{n}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("22a42f"),
    Formula(Equal(BranchCuts(LegendrePolynomial(n, z), z, CC), Set())),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

415911

\# \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = n

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\# \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = n

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Cardinality`	$\# S$	Set cardinality
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("415911"),
    Formula(Equal(Cardinality(Zeros(LegendrePolynomial(n, z), ForElement(z, CC))), n)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

df439e

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) \subset \left(-1, 1\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) \subset \left(-1, 1\right)

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("df439e"),
    Formula(Subset(Zeros(LegendrePolynomial(n, z), ForElement(z, CC)), OpenInterval(-1, 1))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

0745ee

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, x_{n,2}, \ldots, x_{n,n}\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, x_{n,2}, \ldots, x_{n,n}\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("0745ee"),
    Formula(Equal(Zeros(LegendrePolynomial(n, z), ForElement(z, CC)), Set(LegendrePolynomialZero(n, k), For(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

b2d723

P_{n}\!\left(\overline{z}\right) = \overline{P_{n}\!\left(z\right)}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

P_{n}\!\left(\overline{z}\right) = \overline{P_{n}\!\left(z\right)}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Conjugate`	$\overline{z}$	Complex conjugate
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b2d723"),
    Formula(Equal(LegendrePolynomial(n, Conjugate(z)), Conjugate(LegendrePolynomial(n, z)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

ea4754

w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}

TeX:

w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}

Definitions:

Fungrim symbol	Notation	Short description
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`Pow`	${a}^{b}$	Power
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("ea4754"),
    Formula(Equal(GaussLegendreWeight(n, k), Div(2, Mul(Sub(1, Pow(LegendrePolynomialZero(n, k), 2)), Pow(ComplexDerivative(LegendrePolynomial(n, t), For(t, LegendrePolynomialZero(n, k), 1)), 2))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, Range(1, n)))))

47b181

\left|\int_{-1}^{1} f(t) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f(t)\right|

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \rho \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \rho > 1 \;\mathbin{\operatorname{and}}\; f(z) \text{ is holomorphic on } z \in \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right)

References:

L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831

TeX:

\left|\int_{-1}^{1} f(t) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f(t)\right|

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \rho \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \rho > 1 \;\mathbin{\operatorname{and}}\; f(z) \text{ is holomorphic on } z \in \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sum`	$\sum_{n} f(n)$	Sum
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate

Source code for this entry:

Entry(ID("47b181"),
    Formula(Where(LessEqual(Abs(Sub(Integral(f(t), For(t, -1, 1)), Sum(Mul(GaussLegendreWeight(n, k), f(LegendrePolynomialZero(n, k))), For(k, 1, n)))), Div(Mul(64, M), Mul(Mul(15, Sub(1, Pow(rho, -2))), Pow(rho, Mul(2, n))))), Equal(M, Supremum(Abs(f(t)), ForElement(t, BernsteinEllipse(rho)))))),
    Variables(f, n, rho),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(rho, RR), Greater(rho, 1), IsHolomorphic(f(z), ForElement(z, InteriorClosure(BernsteinEllipse(rho)))))),
    References("L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831"))

c8d10e

{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

TeX:

{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("c8d10e"),
    Formula(GreaterEqual(Sub(Pow(Parentheses(LegendrePolynomial(n, x)), 2), Mul(LegendrePolynomial(Sub(n, 1), x), LegendrePolynomial(Add(n, 1), x))), 0)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, ClosedInterval(-1, 1)))))

227d60

{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) > 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(-1, 1\right)

TeX:

{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) > 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(-1, 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("227d60"),
    Formula(Greater(Sub(Pow(Parentheses(LegendrePolynomial(n, x)), 2), Mul(LegendrePolynomial(Sub(n, 1), x), LegendrePolynomial(Add(n, 1), x))), 0)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, OpenInterval(-1, 1)))))

Particular values

Recurrence and functional equations

Generating functions

Rodrigues' formula

Integrals

Sum representations

Hypergeometric representations

Bounds and inequalities

Analytic properties

Gauss-Legendre quadrature

Bounds and inequalities

Turán's inequalities