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1: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

►For asymptotic approximations for the accessory parameter eigenvalues $q_m$, see Fedoryuk (1991) and Slavyanov (1996). ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

2: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

3: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

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4: 8.16 Generalizations

… ►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …

5: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

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18.29.2 $$Q_n(z;a,b,c,d\mid q)\sim \frac{z^n{(a{z}^{-1},b{z}^{-1},c{z}^{-1},d{z}^{-1};q)}_{\mathrm{\infty }}}{{(z^{-2},bc,bd,cd;q)}_{\mathrm{\infty }}},$$ $n\to \mathrm{\infty }$; $z,a,b,c,d,q$ fixed.

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Symbols:

$\sim$: asymptotic equality, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.29 and Ch.18

►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

6: 2 Asymptotic Approximations

Chapter 2 Asymptotic Approximations

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7: 9.14 Incomplete Airy Functions

… ►For information, including asymptotic approximations, computation, and applications, see Levey and Felsen (1969), Constantinides and Marhefka (1993), and Michaeli (1996).

8: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

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24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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9: 25.9 Asymptotic Approximations

§25.9 Asymptotic Approximations

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25.9.1 $$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed

Keywords:

asymptoticapproximation

Source:

Titchmarsh (1986b, (4.12.4), p. 79)

Referenced by:

§25.9

Permalink:

http://dlmf.nist.gov/25.9.E1

Encodings:

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See also:

Annotations for §25.9 and Ch.25

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25.9.3 $$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

asymptoticapproximation

Source:

Titchmarsh (1986b, (4.17.1), p. 88)

Referenced by:

§25.10(ii)

Permalink:

http://dlmf.nist.gov/25.9.E3

Encodings:

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See also:

Annotations for §25.9 and Ch.25

►For other asymptotic approximations see Berry and Keating (1992), Paris and Cang (1997); see also Paris and Kaminski (2001, pp. 380–389).

10: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$. … ►For asymptotic approximations for the zeros of $M_n(nx;\beta ,c)$ in terms of zeros of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … ►For asymptotic approximations to $P_n^{(\lambda )}(x;\varphi )$ as $|x+i\lambda |\to \mathrm{\infty }$, with $n$ fixed, see Temme and López (2001). …Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.