limiting forms for large ℓ

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11: 26.10 Integer Partitions: Other Restrictions

… ►It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). ►

§26.10(v) Limiting Form

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12: 2.8 Differential Equations with a Parameter

… ►Many special functions satisfy an equation of the form …in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. …The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. … ►The transformed equation has the form …where $\rho =lim({\xi }^2\psi (\xi ))$ as $\xi \to 0$. …

13: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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§8.11(i) Large $z$, Fixed $a$

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§8.11(ii) Large $a$, Fixed $z$

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§8.11(iii) Large $a$, Fixed $z/a$

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14: 2.1 Definitions and Elementary Properties

… ►Let $𝐗$ be a point set with a limit point $c$. … ►If $c$ is a finite limit point of $𝐗$, then … ►Asymptotic expansions of the forms (2.1.14), (2.1.16) are unique. … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. …

15: 30.16 Methods of Computation

… ►If $|{\gamma }^2|$ is large we can use the asymptotic expansions in §30.9. … ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements … ►If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. … ►Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to … ►

30.16.8 $$a_{n,k}^m({\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}{e}_{k+p,d},$$

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Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Permalink:

http://dlmf.nist.gov/30.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(ii), §30.16 and Ch.30

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16: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

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External Links:

MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§19.31.

Encodings:

BibTeX

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

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Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.

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Referenced by:

§25.16(i).

Encodings:

BibTeX

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On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

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External Links:

MathReview (T. Hayashida)

Referenced by:

§27.13(ii).

Encodings:

BibTeX

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17: 10.72 Mathematical Applications

… ►The canonical form of differential equation for these problems is given by …where $z$ is a real or complex variable and $u$ is a large real or complex parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

18: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer. … ►

§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). … …

19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ►Let $T$ be the self adjoint extension of a formally self-adjoint differential operator $ℒ$ of the form (1.18.28) on an unbounded interval $X\subset ℝ$, which we will take as $X=[0,+\mathrm{\infty })$, and assume that $q(x)\to 0$ monotonically as $x\to \mathrm{\infty }$, and that the eigenfunctions are non-vanishing but bounded in this same limit. … ►Consider formally self-adjoint operators of the form … ►By Weyl’s alternative $n_1$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_2$. … A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …

20: 2.4 Contour Integrals

… ►Except that $\lambda$ is now permitted to be complex, with $\mathrm{\Re }\lambda >0$, we assume the same conditions on $q(t)$ and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of $\mathrm{\Re }z$. Then … ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►The final expansion then has the form …