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21: 2.7 Differential Equations

… ►Let ${\alpha }_1$, ${\alpha }_2$ denote the indices or exponents, that is, the roots of the indicial equation … ►

2.7.8 $${\mathrm{e}}^{{\lambda }_{j}z}{z}^{{\mu }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{z^s},$$ $j=1,2$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $a_{s,j}$: coefficients, ${\lambda }_j$: roots and ${\mu }_j$: quantities

Referenced by:

§2.11(v), §2.7(ii)

Permalink:

http://dlmf.nist.gov/2.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(ii), §2.7 and Ch.2

►where ${\lambda }_1$, ${\lambda }_2$ are the roots of the characteristic equation … ►

2.7.10 $${\mu }_j=-(f_1{\lambda }_j+g_1)/(f_0+2{\lambda }_j),$$

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Symbols:

${\lambda }_j$: roots, ${\mu }_j$: quantities, $f_s$: coefficients and $g_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(ii), §2.7 and Ch.2

… ►The transformed differential equation either has a regular singularity at $t=\mathrm{\infty }$, or its characteristic equation has unequal roots. …

22: Bibliography R

… ►

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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23: 4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
…

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24: 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

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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

…

25: 23.15 Definitions

… ►In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when $\tau$ lies on the positive imaginary axis the cube root is real and positive. …

26: 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

…

27: 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, $\zeta$. …

28: 14.28 Sums

… ►where the branches of the square roots have their principal values when $z_1,z_2\in (1,\mathrm{\infty })$ and are continuous when $z_1,z_2\in ℂ\setminus (0,1]$. …

29: 22.17 Moduli Outside the Interval [0,1]

… ►In (22.17.5) either value of the square root can be chosen. …

30: 23.18 Modular Transformations

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