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41: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

… ►unless $a=0,-1,\mathrm{\dots }$ and $b-a=0,-1,\mathrm{\dots }$. … ►

§13.7(ii) Error Bounds

… ►

§13.7(iii) Exponentially-Improved Expansion

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

42: 13.29 Methods of Computation

… ►For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. …

43: 28.7 Analytic Continuation of Eigenvalues

… ►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. …To 4D the first branch points between $a_0\left(q\right)$ and $a_2\left(q\right)$ are at $q_0=\pm \mathrm{i}1.4688$ with $a_0\left(q_0\right)=a_2\left(q_0\right)=2.0886$, and between $b_2\left(q\right)$ and $b_4\left(q\right)$ they are at $q_1=\pm \mathrm{i}6.9289$ with $b_2\left(q_1\right)=b_4\left(q_1\right)=11.1904$. … ►All the $a_{2n}\left(q\right)$, $n=0,1,2,\mathrm{\dots }$, can be regarded as belonging to a complete analytic function (in the large). …Analogous statements hold for $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$, and $b_{2n+2}\left(q\right)$, also for $n=0,1,2,\mathrm{\dots }$. …

44: 2.10 Sums and Sequences

… ►

$f(z)$ is real when $a\le z\le n$.

… ►for large $n$. … ►As a first estimate for large $n$ … ►(5.11.7) shows that the integrals around the large quarter circles vanish as $n\to \mathrm{\infty }$. … ►

Example

…

45: 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 $k$ … ►

9.9.10 $$b_k=-T\left(\frac{3}{8}\pi (4k-3)\right),$$

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Symbols:

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

Source:

Olver (1954, (A 21))

Permalink:

http://dlmf.nist.gov/9.9.E10

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). …

46: 25.11 Hurwitz Zeta Function

… ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►

$a$-Derivative

… ►When $a=1$, (25.11.35) reduces to (25.2.3). … ►

§25.11(xii) $a$-Asymptotic Behavior

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

47: 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.

…

48: 28.34 Methods of Computation

… ►

Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

… ►Methods for computing the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$, defined in §§28.2(v) and 28.12(i), include: … ►

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)).

… ►

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

49: 11.9 Lommel Functions

… ►can be regarded as a generalization of (11.2.7). …where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by …and … ►

§11.9(iii) Asymptotic Expansion

… ►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). …

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

ⓘ

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).