DLMF: Untitled Document

… ►For 40D values of the first 500 roots of

\tan x=x

, see Robinson (1972). (These roots are zeros of the Bessel function

J_{3/2}\left(x\right)

; see §10.21.) ►For 10S values of the first five complex roots of

\sin z=az

,

\cos z=az

, and

\cosh z=az

, for selected positive values of

a

, see Fettis (1976). …

… ►Then for

y>f(a)

the equation

f(x)=y

has a unique root

x=x(y)

in

(a,\infty)

, and ►

2.2.2

x(y)\sim y,

y\to\infty

.

ⓘ

Symbols:: $\sim$ : asymptotic equality and $y$ : root
Referenced by:: §2.2
Permalink:: http://dlmf.nist.gov/2.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2 and Ch.2

… ►

2.2.3

t^{2}-\ln t=y.

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.4

t=y^{\frac{1}{2}}\left(1+o\left(1\right)\right),

y\to\infty

.

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.5

t^{2}=y+\ln t=y+\tfrac{1}{2}\ln y+o\left(1\right),

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

…

… ►Roots of

f(z)=0

are

2+\sqrt[3]{4}+\sqrt[3]{2}

,

2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}

,

2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho

. … ►The square roots are chosen so that … ►

§1.11(iv) Roots of Unity and of Other Constants

►The roots of … ►The roots of …

… ►

23.7.1

\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.2

\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.3

\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►where

k,k^{\prime}

and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

… ►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation ►

23.21.1

\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $z$ : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

… ►

23.21.3

f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

►Another form is obtained by identifying

e_{1}

,

e_{2}

,

e_{3}

as lattice roots (§23.3(i)), and setting …

… ►

§23.3(i) Invariants, Roots, and Discriminant

… ►The lattice roots satisfy the cubic equation … ►

23.3.4

\Delta={g_{2}}^{3}-27{g_{3}}^{2}=16(e_{2}-e_{3})^{2}(e_{3}-e_{1})^{2}(e_{1}-e_% {2})^{2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $\Delta$ : discriminant
A&S Ref:: 18.1.8
Referenced by:: §23.19
Permalink:: http://dlmf.nist.gov/23.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.5

e_{1}+e_{2}+e_{3}=0,

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.11
Referenced by:: (23.22.1), §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.7

g_{3}=4e_{1}e_{2}e_{3}=\tfrac{4}{3}({e_{1}}^{3}+{e_{2}}^{3}+{e_{3}}^{3}).

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.10
Referenced by:: (23.22.1), §23.22(ii), §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

…

… ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …

… ►Let

R_{+}

be the set of positive roots and let

\mathbf{v}\mapsto\kappa_{\mathbf{v}}

be a nonnegative function defined on

R_{+}

with the property that it takes constant value in each conjugacy class of roots. … ►

§37.19(vi) OPs Associated with Root Systems

►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials of several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …For general

q

they occur as Macdonald polynomials for root system $A_{n}$ , as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992). See §37.8 and §37.9 for Jacobi polynomials associated with root systems

BC_{2}

and

A_{2}

, respectively.

… ►

2.9.4

\rho_{j}^{n}n^{\alpha_{j}}\sum_{s=0}^{\infty}\frac{a_{s,j}}{n^{s}},

j=1,2

,

ⓘ

Symbols:: $n$ : nonnegative integer, $a_{s,j}$ : coefficients and $\rho_{j}$ : roots
Permalink:: http://dlmf.nist.gov/2.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §2.9(i), §2.9 and Ch.2

►where

\rho_{1},\rho_{2}

are the roots of the characteristic equation … ►When the roots of (2.9.5) are equal we denote them both by

\rho

. Assume first

2g_{1}\neq f_{0}f_{1}

. … ►Then the indices

\alpha_{1},\alpha_{2}

are the roots of …

roots

1—10 of 88 matching pages

1: 4.46 Tables

2: 2.2 Transcendental Equations

3: 1.11 Zeros of Polynomials

§1.11(iv) Roots of Unity and of Other Constants

4: 23.7 Quarter Periods

5: Tom H. Koornwinder

6: 23.21 Physical Applications

7: 23.3 Differential Equations

§23.3(i) Invariants, Roots, and Discriminant

8: 19.38 Approximations

9: 37.19 Other Orthogonal Polynomials of $d$ Variables

§37.19(vi) OPs Associated with Root Systems

10: 2.9 Difference Equations

roots

1—10 of 88 matching pages

1: 4.46 Tables

2: 2.2 Transcendental Equations

3: 1.11 Zeros of Polynomials

§1.11(iv) Roots of Unity and of Other Constants

4: 23.7 Quarter Periods

5: Tom H. Koornwinder

6: 23.21 Physical Applications

7: 23.3 Differential Equations

§23.3(i) Invariants, Roots, and Discriminant

8: 19.38 Approximations

9: 37.19 Other Orthogonal Polynomials of d Variables

§37.19(vi) OPs Associated with Root Systems

10: 2.9 Difference Equations

9: 37.19 Other Orthogonal Polynomials of $d$ Variables