asymptotic expansions for large zeros

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11: 13.9 Zeros

… ► … ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by … … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $U(a,b,z)$ are given by …

12: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

►

§10.19(i) Asymptotic Forms

… ►

§10.19(ii) Debye’s Expansions

… ►

§10.19(iii) Transition Region

… ►See also §10.20(i).

13: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. …If $m$ is a large positive integer, then ►

$$\text{<span class="ltx\_text hilite">zeros</span> of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

… ►In the case $\nu =0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …

14: 10.18 Modulus and Phase Functions

… ►

§10.18(iii) Asymptotic Expansions for Large Argument

… ►

10.18.19 $$N_{\nu }^2\left(x\right)\sim \frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu -3}{{(2x)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2x)}^4}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.30 (corrected)

Referenced by:

§10.18(iii), §10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.18.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►the general term in this expansion being … ►

10.18.21 $${\varphi }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu -\frac{1}{4}\right)\pi +\frac{\mu +3}{2(4x)}+\frac{{\mu }^2+46\mu -63}{6{(4x)}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(4x)}^5}+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.31

Referenced by:

§10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

15: 28.34 Methods of Computation

… ►

Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

… ►

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

… ►

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

… ►

Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

… ►

Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

16: 6.18 Methods of Computation

… ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. … ►

§6.18(iii) Zeros

►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

17: Bibliography N

… ►

The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

… ►

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

Encodings:

BibTeX

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

…

18: Bibliography F

… ►

On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:

ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

… ►

On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

…

19: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►

§29.20(iii) Zeros

… ►Alternatively, the zeros can be found by locating the maximum of function $g$ in (29.12.11).

20: 2.8 Differential Equations with a Parameter

… ►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. …The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. Zeros of $f(z)$ are also called turning points. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►However, in all cases with $\lambda >-2$ and $\lambda \ne 0$ or $\pm 1$, only uniform asymptotic approximations are available, not uniform asymptotic expansions. …