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51: 2.10 Sums and Sequences

… ►As a first estimate for large $n$ …

52: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►If $k$ is regarded as a continuous variable, then … ►For large $k$ ►

9.9.6 $$a_k=-T\left(\frac{3}{8}\pi (4k-1)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

53: 25.11 Hurwitz Zeta Function

… ►

§25.11(xii) $a$-Asymptotic Behavior

… ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

54: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

…

55: 11.9 Lommel Functions

… ►can be regarded as a generalization of (11.2.7). …and … ►

§11.9(iii) Asymptotic Expansion

… ►For $a_k(\mu ,\nu )$ see (11.9.4). … ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …

56: 28.35 Tables

… ►

National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

…

57: 16.13 Appell Functions

… ►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

… ►

16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

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Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and ${\gamma }^{\prime }$ is a nonpositive integer. … ►For large parameter asymptotics see López et al. (2013a, b), and Ferreira et al. (2013a, b).

58: 2.1 Definitions and Elementary Properties

… ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). … ►If $\sum a_s{x}^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to \mathrm{\infty }$ in $ℂ$. …

59: 27.13 Functions

… ►Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. …

60: 28.34 Methods of Computation

… ►

Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

… ►

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

… ►

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.