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1: 4.46 Tables

… ►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). …

2: 2.2 Transcendental Equations

… ►Then for $y>f(a)$ the equation $f(x)=y$ has a unique root $x=x(y)$ in $(a,\mathrm{\infty })$, and ►

2.2.2 $$x(y)\sim y,$$ $y\to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality and $y$: root

Referenced by:

§2.2

Permalink:

http://dlmf.nist.gov/2.2.E2

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Annotations for §2.2 and Ch.2

… ►

2.2.3 $$t^2-\ln t=y.$$

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Symbols:

$\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E3

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Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.4 $$t=y^{\frac{1}{2}}\left(1+o\left(1\right)\right),$$ $y\to \mathrm{\infty }$.

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Symbols:

$o\left(x\right)$: order less than and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E4

Encodings:

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Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.5 $$t^2=y+\ln t=y+\frac{1}{2}\ln y+o\left(1\right),$$

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Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E5

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Annotations for §2.2, §2.2 and Ch.2

…

3: 1.11 Zeros of Polynomials

… ►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 …

4: 23.7 Quarter Periods

… ►

23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: 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

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Annotations for §23.7 and Ch.23

►

23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: 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:

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Annotations for §23.7 and Ch.23

►

23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: 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

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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.

5: Tom H. Koornwinder

… ►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. …

6: 23.21 Physical Applications

… ►Ellipsoidal coordinates $(\xi ,\eta ,\zeta )$ may be defined as the three roots $\rho$ of the equation ►

23.21.1 $$\frac{x^2}{\rho -e_1}+\frac{y^2}{\rho -e_2}+\frac{z^2}{\rho -e_3}=1,$$

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Symbols:

$e_j$: Weierstrass lattice roots, $𝕃$: lattice, $x$: real part of $z$ , $y$: imaginary part of $z$ , $z$: complex and $\rho$: roots

Permalink:

http://dlmf.nist.gov/23.21.E1

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Annotations for §23.21(iii), §23.21 and Ch.23

… ►

23.21.3 $$f(\rho )=2{\left((\rho -e_1)(\rho -e_2)(\rho -e_3)\right)}^{1/2}.$$

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Symbols:

$e_j$: Weierstrass lattice roots, $𝕃$: lattice, $\rho$: roots and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E3

Encodings:

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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 …

7: 23.3 Differential Equations

… ►

§23.3(i) Invariants, Roots, and Discriminant

… ►The lattice roots satisfy the cubic equation … ►

23.3.4 $$\mathrm{\Delta }=g_2^3-27g_3^2=16{(e_2-e_3)}^2{(e_3-e_1)}^2{(e_1-e_2)}^2.$$

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Symbols:

$e_j$: Weierstrass lattice roots, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\mathrm{\Delta }$: discriminant

A&S Ref:

18.1.8

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.3.E4

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Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.5 $$e_1+e_2+e_3=0,$$

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Symbols:

$e_j$: Weierstrass lattice roots and $𝕃$: lattice

A&S Ref:

18.1.11

Referenced by:

(23.22.1), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.3.E5

Encodings:

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Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.7 $$g_3=4e_1{e}_2{e}_3=\frac{4}{3}(e_1^3+e_2^3+e_3^3).$$

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Symbols:

$e_j$: Weierstrass lattice roots, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ and $𝕃$: 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:

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Annotations for §23.3(i), §23.3 and Ch.23

…

8: 19.38 Approximations

… ►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). …

9: 2.9 Difference Equations

… ►

2.9.4 $${\rho }_j^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $j=1,2$,

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Symbols:

$n$: nonnegative integer, $a_{s,j}$: coefficients and ${\rho }_j$: roots

Permalink:

http://dlmf.nist.gov/2.9.E4

Encodings:

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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 $2g_1\ne f_0{f}_1$. … ►Then the indices ${\alpha }_1,{\alpha }_2$ are the roots of …

10: 4.43 Cubic Equations

… ►The roots of … ►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …