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21: 10.14 Inequalities; Monotonicity

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10.14.9 $$|J_n\left(nz\right)|\le 1,$$ $n=0,1,2,\mathrm{\dots },z\in 𝐊$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $n$: integer, $z$: complex variable and $𝐊$: domain

Permalink:

http://dlmf.nist.gov/10.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14, §10.14 and Ch.10

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22: Bibliography R

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Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-5159-4, MathReview, ZentralBlatt

Referenced by:

Ch.35, Profile Donald St. P. Richards.

Encodings:

BibTeX

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23: 2.8 Differential Equations with a Parameter

… ►in which $\xi$ ranges over a bounded or unbounded interval or domain $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. … ►

2.8.11 $$W_{n,1}(u,\xi )={\mathrm{e}}^{u\xi }\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^s}+O\left(\frac{1}{u^n}\right)\right),$$ $\xi \in {𝚫}_1({\alpha }_1)$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter

Referenced by:

§2.8(ii), §2.8(ii)

Permalink:

http://dlmf.nist.gov/2.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(ii), §2.8 and Ch.2

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2.8.12 $$W_{n,2}(u,\xi )={\mathrm{e}}^{-u\xi }\left(\sum _{s=0}^{n-1}{(-1)}^s\frac{A_s(\xi )}{u^s}+O\left(\frac{1}{u^n}\right)\right),$$ $\xi \in {𝚫}_2({\alpha }_2)$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter

Referenced by:

§2.8(ii), §2.8(ii)

Permalink:

http://dlmf.nist.gov/2.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(ii), §2.8 and Ch.2

… ►The regions of validity ${𝚫}_j({\alpha }_j)$ comprise those points $\xi$ that can be joined to ${\alpha }_j$ in $𝚫$ by a path ${𝒬}_j$ along which $\mathrm{\Re }v$ is nondecreasing $(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from ${\alpha }_j$ to $\xi$. …

24: Bibliography

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Arb: A C Library for Arbitrary Precision Ball Arithmetic.

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Notes:

An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Chiral-symmetry breaking in QCD. I. The infrared domain. Phys. Rev. D (3) 37 (8), pp. 2290–2295.

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External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

§15.18.

Encodings:

BibTeX

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25: Bibliography F

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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:

MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

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26: Bibliography W

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.

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External Links:

ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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27: 3.10 Continued Fractions

… ►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5). …

28: 31.11 Expansions in Series of Hypergeometric Functions

… ►The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. …

29: 4.23 Inverse Trigonometric Functions

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30: 18.15 Asymptotic Approximations

… ►For large $\beta$, fixed $\alpha$, and $0\le n/\beta \le c$, Dunster (1999) gives asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ that are uniform in unbounded complex $z$-domains containing $z=\pm 1$. …The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. …