Dunster’s

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1: 28.8 Asymptotic Expansions for Large $q$

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Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►

2: Bibliography D

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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5 (3), pp. 223–247.

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External Links:

ISSN 1073-2772, Pdf, MathReview (John G. Stalker), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

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3: 8.22 Mathematical Applications

… ►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

4: 10.49 Explicit Formulas

… ►See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c). … ►

§10.49(iii) Rayleigh’s Formulas

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10.49.17 $$s_k(n+\frac{1}{2})=\frac{(2k)!(n+k)!}{2^{2k}{(k!)}^2(n-k)!},$$ $k=0,1,\mathrm{\dots },n$.

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Defines:

$s_k(n)$ (locally)

Symbols:

$!$: factorial (as in $n!$), $n$: integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.49.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(iv), §10.49 and Ch.10

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10.49.18 $${𝗃}_n^2\left(z\right)+{𝗒}_n^2\left(z\right)=\sum _{k=0}^n\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$

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Symbols:

${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $s_k(n)$

A&S Ref:

10.1.27

Referenced by:

§10.49(iv)

Permalink:

http://dlmf.nist.gov/10.49.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(iv), §10.49 and Ch.10

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10.49.20 $${\left({𝗂}_n^{(1)}\left(z\right)\right)}^2-{\left({𝗂}_n^{(2)}\left(z\right)\right)}^2={(-1)}^{n+1}\sum _{k=0}^n{(-1)}^k\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$

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Symbols:

${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $s_k(n)$

A&S Ref:

10.2.26

Referenced by:

§10.49(iv)

Permalink:

http://dlmf.nist.gov/10.49.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(iv), §10.49 and Ch.10

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5: Staff

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Howard S. Cohl, Technical Editor, NIST

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T. Mark Dunster, San Diego State University, Chap. 14

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Roderick S. C. Wong, City University of Hong Kong, Chaps. 1, 2, 18

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T. Mark Dunster, San Diego State University, for Chap. 14

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Roderick S. C. Wong, City University of Hong Kong, for Chaps. 2, 18

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6: Preface

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T. Mark Dunster, San Diego State University

… ►Boggs, S. … S. … S. … S. …

7: 2.8 Differential Equations with a Parameter

… ►For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. ►For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …

8: 2.11 Remainder Terms; Stokes Phenomenon

… ►with … ►Application of Watson’s lemma (§2.4(i)) yields … ►See also Paris and Kaminski (2001, Chapter 6), and Dunster (1996b, 1997). … ►For error bounds see Dunster (1996c). … ► …

9: 18.15 Asymptotic Approximations

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18.15.1 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-1}{2}^{2n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1)\left(\sum _{m=0}^{M-1}\frac{f_m(\theta )}{2^m{\left(2n+\alpha +\beta +2\right)}_m}+O\left(n^{-M}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $f_m(\theta )$

Referenced by:

§18.15(i), §18.15(i)

Permalink:

http://dlmf.nist.gov/18.15.E1

Encodings:

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See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.3 $$C_{m,\mathrm{\ell }}(\alpha ,\beta )={\left(\frac{1}{2}+\alpha \right)}_{\mathrm{\ell }}{\left(\frac{1}{2}-\alpha \right)}_{\mathrm{\ell }}{\left(\frac{1}{2}+\beta \right)}_{m-\mathrm{\ell }}{\left(\frac{1}{2}-\beta \right)}_{m-\mathrm{\ell }},$$

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Defines:

$C_{m,\mathrm{\ell }}(\alpha ,\beta )$ (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\ell }$: nonnegative integer and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►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 asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

10: 14.15 Uniform Asymptotic Approximations

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14.15.3 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). … ►For convergent series expansions see Dunster (2004). … ►For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986). …