asymptotic approximations for large order

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11: 25.11 Hurwitz Zeta Function

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§25.11(xii) $a$-Asymptotic Behavior

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25.11.41 $$\zeta (s,a+1)=\zeta \left(s\right)-s\zeta \left(s+1\right)a+O\left(a^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptoticapproximation

Source:

Apostol (1952, (15), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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25.11.42 $$\zeta (s,\alpha +\mathrm{i}\beta )\to 0,$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{i}$: imaginary unit and $s$: complex variable

Keywords:

asymptoticapproximation

Source:

Apostol (1952, (16), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

12: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

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§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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13: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

… ► ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►

§2.11(iii) Exponentially-Improved Expansions

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§2.11(vi) Direct Numerical Transformations

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14: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

… ► … ►For uniform approximations valid when $\mu$ is large, $x/\mathrm{i}\in (0,\mathrm{\infty })$, and $\kappa /\mathrm{i}\in [0,\mu /\delta ]$, see Olver (1997b, pp. 401–403). … ►

15: 13.22 Zeros

… ►Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by ►

13.22.1 $${\varphi }_r=\frac{j_{2\mu ,r}^2}{4\kappa }+j_{2\mu ,r}O\left({\kappa }^{-\frac{3}{2}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\varphi }_r$: positive zeros and $j_{b,r}$: positive zero of Bessel

Referenced by:

§13.22, §13.22

Permalink:

http://dlmf.nist.gov/13.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.22 and Ch.13

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16: 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). For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …

17: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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External Links:

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

Referenced by:

§30.9(i).

Encodings:

BibTeX

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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External Links:

ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

Encodings:

BibTeX

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Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

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External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

BibTeX

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.22(ii).

Encodings:

BibTeX

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18: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: … ►For the asymptotic behavior of $F(\varphi ,k)$ and $E(\varphi ,k)$ as $\varphi \to \frac{1}{2}\pi -$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. …

19: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►These approximations are proved in Dunster (1989). … ►These approximations are proved in Dunster (1989). … ►

§13.21(iv) Large $\kappa$, Other Expansions

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20: 13.9 Zeros

… ► … ►where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … ►where $n$ is a large positive integer. … ►where $n$ is a large positive integer. …