expansion of arbitrary vector

SearchAdvancedHelp

(0.001 seconds)

11—20 of 351 matching pages

11: 2.1 Definitions and Elementary Properties

… ►

§2.1(iii) Asymptotic Expansions

… ►Symbolically, … ►For an example see (2.8.15). … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►

§2.1(v) Generalized Asymptotic Expansions

…

12: 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).

13: Bibliography L

… ►

A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

… ►

Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Walter Schempp), ZentralBlatt

Referenced by:

§2.6(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

ⓘ

External Links:

Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

Encodings:

BibTeX

… ►

VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.

ⓘ

Notes:

Single-precision Fortran, minimum accuracy 13D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

…

14: 13.31 Approximations

… ►

§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. … ►

13.31.1 $$A_n(z)=\sum _{s=0}^n\frac{{\left(-n\right)}_s{\left(n+1\right)}_s{\left(a\right)}_s{\left(b\right)}_s}{{\left(a+1\right)}_s{\left(b+1\right)}_s{(n!)}^2}{}_3{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-n+s,n+1+s,1}{1+s,a+1+s,b+1+s};-z),$$

ⓘ

Defines:

$A_n(z)$: expansion (locally)

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

… ►

13.31.2 $$B_n(z)={}_2{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+1}{a+1,b+1};-z).$$

ⓘ

Defines:

$B_n(z)$: expansion (locally)

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

… ►

13.31.3 $$z^{a}U(a,1+a-b,z)=\underset{n\to \mathrm{\infty }}{lim}\frac{A_n(z)}{B_n(z)}.$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer, $z$: complex variable, $A_n(z)$: expansion and $B_n(z)$: expansion

Permalink:

http://dlmf.nist.gov/13.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

15: 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11 Asymptotic Expansions of Anger–Weber Functions

… ► … ►where … ►and … ►For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020). …

16: Bibliography O

… ►

Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

BibTeX

… ►

Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

►

Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

… ►

On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

… ►

Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

…

17: Bibliography S

… ►

Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

ⓘ

External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.

ⓘ

External Links:

ISSN 0025-5718, Document, Link, MathReview (Vittoria Demichelis), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

►

Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36 (3), pp. 331–352.

ⓘ

External Links:

ISSN 0176-4276, Document, Link, MathReview (Razvan A. Mezei), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

… ►

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

… ►

Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

ⓘ

External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

…

18: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

… ►For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955). … ►

§33.12(ii) Uniform Expansions

… ►The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. … ►

19: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►

28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when ►

28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►where $\delta$ again denotes an arbitrary small positive constant. …

20: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

►

§10.17(i) Hankel’s Expansions

… ► ►

§10.17(ii) Asymptotic Expansions of Derivatives

… ►

§10.17(v) Exponentially-Improved Expansions

…