roots of constants

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21: Errata

… ►

Section 4.43

The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

Let $p$ $(\neq 0)$ and $q$ be real constants and

4.43.1

A=\left(-\tfrac{4}{3}p\right)^{1/2},

B=\left(\tfrac{4}{3}p\right)^{1/2}.

The roots of

4.43.2

z^{3}+pz+q=0

are:

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=4q/A^{3}$ , when $4p^{3}+27q^{2}\leq 0$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$ , with $\cosh\left(3a\right)=-4q/A^{3}$ , when $p<0$ , $q<0$ , and $4p^{3}+27q^{2}>0$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

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. See also §1.11(iii).

Reported 2014-10-31 by Masataka Urago.

…

22: 10.9 Integral Representations

… ►where

\gamma

is Euler’s constant (§5.2(ii)). … ►where

c

is a positive constant and the integration path encloses the points

t=0,-1,-2,\dotsc

. ►In (10.9.24) and (10.9.25)

c

is any constant exceeding

\max(\Re\nu,0)

. … ►where the square root has its principal value. …where

c

is a positive constant. …

23: 19.31 Probability Distributions

… ►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. … ►

19.31.2

\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),

\mu>-\tfrac{1}{2}n

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\mathbb{R}$ : real line, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $n$ : nonnegative integer and $\lambda_{n}$ : eigenvalues
Referenced by:: §19.31
Permalink:: http://dlmf.nist.gov/19.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

…

24: 18.38 Mathematical Applications

… ►For applications of Krawtchouk polynomials

K_{n}\left(x;p,N\right)

and

q

-Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)

to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). … ►where

Q_{0}

is a constant with explicit expression in terms of

e_{1},e_{2},e_{3},e_{4}

and

q

given in Koornwinder (2007a, (2.8)). ►The abstract associative algebra with generators

K_{0},K_{1},K_{2}

and relations (18.38.4), (18.38.6) and with the constants

B,C_{0},C_{1},D_{0},D_{1}

in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). … ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. …

25: 31.11 Expansions in Series of Hypergeometric Functions

… ►In this case the accessory parameter

q

is a root of the continued-fraction equation … ►

\lambda=\gamma+\delta-1,

… ►

\lambda=\gamma,

… ►

\mu=\gamma-1,

… ►

\mu=\gamma+\delta-2.

…

26: 19.14 Reduction of General Elliptic Integrals

… ►In (19.14.4)

0\leq y<x

, each quadratic polynomial is positive on the interval

(y,x)

, and

\alpha,\beta,\gamma

is a permutation of

0,a_{1}b_{2},a_{2}b_{1}

(not all 0 by assumption) such that

\alpha\leq\beta\leq\gamma

. … ►

19.14.5

{\sin}^{2}\phi=\frac{\gamma-\alpha}{U^{2}+\gamma},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.7

{\sin}^{2}\phi=\frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}.

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.8

{\sin}^{2}\phi=\frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}.

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

… ►It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …

27: Bibliography S

… ►

H. Sakai (2001) Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220 (1), pp. 165–229.

ⓘ

External Links:: ISSN 0010-3616, Document, MathReview (I. Dolgachev), ZentralBlatt
Referenced by:: §32.7(viii).
Encodings:: BibTeX

… ►

A. J. Stone and C. P. Wood (1980) Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 16D
External Links:: Document, Code
Referenced by:: 10th item.
Encodings:: BibTeX

… ►

J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941) Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

ⓘ

External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §28.1, 7th item, Notations S, Notations S.
Encodings:: BibTeX

►

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (1956) Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.

ⓘ

External Links:: ISBN 0262190028, 0262692910, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: 1st item, Ch.30.
Encodings:: BibTeX

… ►

C. E. Synolakis (1988) On the roots of

f(z)=J_{0}(z)-iJ_{1}(z)

. Quart. Appl. Math. 46 (1), pp. 105–107.

ⓘ

External Links:: ISSN 0033-569X, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

…

28: 19.20 Special Cases

… ►

R_{F}\left(0,0,z\right)=\infty.

►The first lemniscate constant is given by ►

19.20.2

\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)=\frac% {\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;8777% 1\;46059\;90523\;\dots.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Notes:: For more digits see OEIS Sequence A085565; see also Sloane (2003).
Referenced by:: §19.20(i), §19.20(iv), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.20.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(i), §19.20 and Ch.19

… ►

19.20.19

R_{D}\left(x,y,z\right)\sim 3x^{-1/2}y^{-1/2}z^{-1/2},

z/\sqrt{xy}\to 0

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $\sim$ : asymptotic equality
Referenced by:: §19.20(iv), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.20.E19
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

… ►The second lemniscate constant is given by …

29: 10.25 Definitions

… ►

10.25.2

I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}% {4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.

ⓘ

Defines:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.10
Referenced by:: §10.30(i), §10.31, §10.38, §10.45, §10.46, §10.53, §8.6(i)
Permalink:: http://dlmf.nist.gov/10.25.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.25(ii), §10.25 and Ch.10

… ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. …

30: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►where the branch of the square root is continuous and satisfies

\eta(\lambda)\sim\lambda-1

\lambda\to 1

. … ►

8.12.5

\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ze^{\pm\pi i}\right)=% \pm\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)-iT(a,\eta),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $T(a,\eta)$ : function
Referenced by:: Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)
Permalink:: http://dlmf.nist.gov/8.12.E5
Encodings:: pMML, png, TeX
Change (effective with 1.0.16):: To be consistent with the notation used in (8.12.16), the original equation,
$\Gamma\left(a+1\right)\frac{e^{\pm\pi ia}}{2\pi i}\Gamma\left(-a,ze^{\pm\pi i}% \right)=\mp\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)+iT(% a,\eta),$
was rewritten.
See also:: Annotations for §8.12 and Ch.8

… ►

8.12.6

z^{-a}\gamma^{*}\left(-a,-z\right)=\cos\left(\pi a\right)-2\sin\left(\pi a% \right)\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}F\left(\eta\sqrt{a/2}% \right)+T(a,\eta)\right),

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $T(a,\eta)$ : function
Permalink:: http://dlmf.nist.gov/8.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §8.12 and Ch.8

… ►where

g_{k}

k=0,1,2,\dots

, are the coefficients that appear in the asymptotic expansion (5.11.3) of

\Gamma\left(z\right)

. … ►Lastly, a uniform approximation for

\Gamma\left(a,ax\right)

for large

a

, with error bounds, can be found in Dunster (1996a). …