asymptotic expansions for large q

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21: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►The asymptotic expansion of the large positive zeros (not necessarily the $m$th) of the function …

22: 14.15 Uniform Asymptotic Approximations

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§14.15(i) Large $\mu$, Fixed $\nu$

… ►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). ►

§14.15(iii) Large $\nu$, Fixed $\mu$

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23: 14.20 Conical (or Mehler) Functions

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§14.20(vii) Asymptotic Approximations: Large $\tau$, Fixed $\mu$

… ►For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … ►

§14.20(viii) Asymptotic Approximations: Large $\tau$, $0\le \mu \le A\tau$

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§14.20(ix) Asymptotic Approximations: Large $\mu$, $0\le \tau \le A\mu$

… ►For extensions to complex arguments (including the range ), asymptotic expansions, and explicit error bounds, see Dunster (1991). …

24: Bibliography W

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

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Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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External Links:

ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

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Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:

ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

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25: Bibliography O

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

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External Links:

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

Referenced by:

§5.18(ii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

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

Referenced by:

§15.12(iii).

Encodings:

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Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:

Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

Encodings:

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26: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§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). … ►

12.11.7 $$u_{a,s}^{\prime }\sim 2^{\frac{1}{2}}\mu \left(q_0(\beta )+\frac{q_1(\beta )}{{\mu }^4}+\frac{q_2(\beta )}{{\mu }^8}+\mathrm{\cdots }\right),$$

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Defines:

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

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter and $\beta$

Permalink:

http://dlmf.nist.gov/12.11.E7

Encodings:

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See also:

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

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27: Bibliography P

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

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Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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External Links:

ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§10.46.

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The asymptotic expansion of a generalised incomplete gamma function. J. Comput. Appl. Math. 151 (2), pp. 297–306.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§8.16.

Encodings:

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

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28: 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. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … ►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. …

29: 27.14 Unrestricted Partitions

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27.14.2 $$f\left(x\right)=\prod _{m=1}^{\mathrm{\infty }}(1-x^m)={(x;x)}_{\mathrm{\infty }},$$ ,

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Defines:

$f\left(x\right)$: Euler’s reciprocal function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $m$: positive integer and $x$: real number

Referenced by:

§27.14(iv), §27.21, Erratum (V1.0.28) for Equation (27.14.2)

Permalink:

http://dlmf.nist.gov/27.14.E2

Encodings:

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Addition (effective with 1.0.28):

The representation in terms of ${(x;x)}_{\mathrm{\infty }}$ was added to this equation.

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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§27.14(iii) Asymptotic Formulas

… ►For large $n$ …Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for $p\left(n\right)$: … ►Ono proved that for every prime $q>3$ there are integers $a$ and $b$ such that $p\left(an+b\right)\equiv 0(modq)$ for all $n$. …

30: Errata

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8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right)$$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

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The symbol $\sim$ is used for two purposes in the DLMF, in some cases for asymptotic equality and in other cases for asymptotic expansion, but links to the appropriate definitions were not provided. In this release changes have been made to provide these links.

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A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

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Because (2.11.4) is not an asymptotic expansion, the symbol $\sim$ that was used originally is incorrect and has been replaced with $\approx$, together with a slight change of wording.

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Originally was expressed in term of asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

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