Neumann%20expansion

SearchAdvancedHelp

(0.001 seconds)

11—20 of 343 matching pages

11: 10.75 Tables

… ►

Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

… ►

Olver (1960) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(\frac{1}{2})20⁤\frac{1}{2}$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$; see §10.21(viii), and more fully Olver (1954).

… ►

Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

… ►

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

… ►

Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

…

12: 12.11 Zeros

… ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … ►

12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

…

13: Bibliography N

… ►

On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

►

On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

ⓘ

External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

►

On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

… ►

Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

… ►

A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

ⓘ

Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

…

14: 2.11 Remainder Terms; Stokes Phenomenon

… ► … ►

§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ► … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

15: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

16: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

►

§5.11(i) Poincaré-Type Expansions

… ►and … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►

17: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►

§28.8(ii) Sips’ Expansions

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). ►

§28.8(iii) Goldstein’s Expansions

…

18: 9.9 Zeros

… ►

§9.9(iv) Asymptotic Expansions

… ►

9.9.6 $$a_k=-T\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.7 $${\mathrm{Ai}}^{\prime }\left(a_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.8 $$a_k^{\prime }=-U\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

19: Bibliography F

… ►

ⓘ

Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

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

Encodings:

BibTeX

… ►

Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

… ►

Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

… ►

Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

ⓘ

External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

Encodings:

BibTeX

… ►

On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

…

20: 6.16 Mathematical Applications

… ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify