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11: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►Solutions are called roots of the equation, or zeros of $f$. … ►Let $z_1,z_2,\mathrm{\dots }$ be a sequence of approximations to a root, or fixed point, $\zeta$. … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

12: 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). …

13: Bibliography M

… ►

Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.

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External Links:

Document, MathReview (D.-N. Verma), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

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Some conjectures for root systems. SIAM J. Math. Anal. 13 (6), pp. 988–1007.

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External Links:

ISSN 0036-1410, Document, MathReview (S. Milne), ZentralBlatt

Referenced by:

§17.13, §17.14.

Encodings:

BibTeX

… ►

Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

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External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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External Links:

ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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Numerical Methods for Roots of Polynomials. Part I. Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.

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External Links:

ISBN 978-0-444-52729-5; 0-444-52729-X, MathReview Entry, ZentralBlatt

Referenced by:

§3.8(iv).

Encodings:

BibTeX

…

14: Bibliography R

… ►

On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

… ►

Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

Encodings:

BibTeX

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Roots of $\tan x=x$ .

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Notes:

10 typewritten pages deposited in the UMT file at Lawrence Berkeley Laboratory, University of California, Berkeley, CA.

Referenced by:

§4.46.

Encodings:

BibTeX

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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15: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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16: 23.9 Laurent and Other Power Series

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$c_2={\displaystyle \frac{1}{20}}{g}_2,$

… ►

23.9.6 $$\mathrm{\wp }\left({\omega }_j+t\right)=e_j+(3{e_j}^2-5c_2)t^2+(10c_2{e}_j+21c_3)t^4+(7c_2{e_j}^2+21c_3{e}_j+5c_2^2)t^6+O\left(t^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $c_n$ and $t$: variable

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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17: 8 Incomplete Gamma and Related
Functions

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18: 28 Mathieu Functions and Hill’s Equation

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19: Bibliography K

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The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

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External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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20: 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

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2 and Ch.2

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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

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

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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:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

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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

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

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