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11: 28.6 Expansions for Small $q$

… ►Numerical values of the radii of convergence ${\rho }_n^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\mathrm{\dots },9$ are given in Table 28.6.1. … ►

Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.

► ►►

$n$	${\rho }_n^{(1)}$		${\rho }_n^{(2)}$		${\rho }_n^{(3)}$
…

►

►It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). … ►

28.6.20 $$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{{\rho }_n^{(j)}}{n^2}\ge k{k}^{\prime }{(K\left(k\right))}^2=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots },$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root

Referenced by:

§28.6(i), §28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E20

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See also:

Annotations for §28.6(i), §28.6 and Ch.28

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12: 14.20 Conical (or Mehler) Functions

… ►

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

… ►

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

… ►

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

… ►and the variable $\rho$ is defined by … ►The interval is mapped one-to-one to the interval , with the points $x=-1$, $x=0$, and $x=1$ corresponding to $\rho =-\mathrm{\infty }$, $\rho =0$, and $\rho =\mathrm{\infty }$, respectively. …

13: 14.15 Uniform Asymptotic Approximations

… ►

§14.15(i) Large $\mu$, Fixed $\nu$

… ►and … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►

§14.15(iii) Large $\nu$, Fixed $\mu$

… ►

14: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

… ►

§13.7(ii) Error Bounds

… ►

$\rho =\frac{1}{2}\left|2a^2-2ab+b\right|+{\displaystyle \frac{\sigma (1+\frac{1}{4}\sigma )}{{(1-\sigma )}^2}}$ ,

… ►

§13.7(iii) Exponentially-Improved Expansion

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

15: 18.39 Applications in the Physical Sciences

… ►Now use spherical coordinates (1.5.16) with $r$ instead of $\rho$, and assume the potential $V$ to be radial. … ►with ${\rho }_n$ and ${ϵ}_n$ being those of (18.39.35), are then … ►

18.39.39 $${𝐋}_p^0(\rho )={\mathrm{e}}^{\rho }\frac{d^p}{{d\rho }^p}({\rho }^p{\mathrm{e}}^{-\rho }),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $\mathrm{e}$: base of natural logarithm

Permalink:

http://dlmf.nist.gov/18.39.E39

Encodings:

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Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. … ►A relativistic treatment becoming necessary as $Z$ becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ${\left(\alpha Z\right)}^2\sim {\left(Z/137\right)}^2$, $\alpha$ being the dimensionless fine structure constant $e^2/(4\pi {\epsilon }_0\mathrm{\hslash }c)$, where $c$ is the speed of light. …

16: 18.15 Asymptotic Approximations

… ►

18.15.5 $$\rho =n+\frac{1}{2}(\alpha +\beta +1).$$

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Defines:

$\rho$ (locally)

Symbols:

$n$: nonnegative integer

Referenced by:

§18.16(ii)

Permalink:

http://dlmf.nist.gov/18.15.E5

Encodings:

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See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►

18.15.7

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Defines:

${\epsilon }_M(\rho ,\theta )$ (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\delta$: arbitrary small positive constant and $\rho$

Permalink:

http://dlmf.nist.gov/18.15.E7

Encodings:

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See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►For large $\beta$, fixed $\alpha$, and $0\le n/\beta \le c$, Dunster (1999) gives asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ that are uniform in unbounded complex $z$-domains containing $z=\pm 1$. …This reference also supplies asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ for large $n$, fixed $\alpha$, and $0\le \beta /n\le c$. … ►These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

17: 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.

… ►

Blanch and Clemm (1969) includes eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $q=\rho {\mathrm{e}}^{\mathrm{i}\varphi }$, $\rho =0(.5)25$, $\varphi =5^{\circ }(5^{\circ }){90}^{\circ }$, $n=0(1)15$; 4D. Also $a_n\left(q\right)$ and $b_n\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho =0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

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18: 2.6 Distributional Methods

… ►To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form … ►

2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

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Annotations for §2.6(ii), §2.6 and Ch.2

… ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. …

19: 28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. … ►

28.33.1 $$\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}-\frac{\rho }{\tau }\frac{{\partial }^{2}W}{{\partial t}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $\rho$: mass, $\tau$: radial tension and $W(x,y,t)$: solution

Permalink:

http://dlmf.nist.gov/28.33.E1

Encodings:

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See also:

Annotations for §28.33(ii), §28.33 and Ch.28

►with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^2={\omega }^2\rho /\tau$. …If we denote the positive solutions $q$ of (28.33.3) by $q_{n,m}$, then the vibration of the membrane is given by ${\omega }_{n,m}^2=4q_{n,m}\tau /(c^2\rho )$. … ►In particular, the equation is stable for all sufficiently large values of $\omega$. …

20: 2.5 Mellin Transform Methods

… ►Let $f(t)$ be a locally integrable function on $(0,\mathrm{\infty })$, that is, ${\int }_{\rho }^{T}f(t)dt$ exists for all $\rho$ and $T$ satisfying . … ►Similarly, if $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$. … ►where $J_{\nu }$ denotes the Bessel function (§10.2(ii)), and $x$ is a large positive parameter. … ►for any $\rho$ satisfying . Similarly, since $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho$ so that $ℳh_2\left(z\right)$ has no poles in . …