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11: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

12: 29.20 Methods of Computation

… ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …

13: 2.8 Differential Equations with a Parameter

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2.8.1 $$d^{2}w/{dz}^2=\left(u^{2}f(z)+g(z)\right)w,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $g(x)$: function, $w$: solution and $u$: large real or complex parameter

Referenced by:

§2.8(i), §2.8(v)

Permalink:

http://dlmf.nist.gov/2.8.E1

Encodings:

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Annotations for §2.8(i), §2.8 and Ch.2

►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 $ℂ$. For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►

2.8.3 $$\frac{d^{2}W}{{d\xi }^2}=\left(u^2{\dot{z}}^{2}f(z)+\psi (\xi )\right)W,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $f(x)$: function, $u$: large real or complex parameter, $W$: change of variable, $\dot{z}$: derivative of $z$ with respect to $\xi$ and $\psi (\xi )$: function

Permalink:

http://dlmf.nist.gov/2.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(i), §2.8 and Ch.2

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2.8.8 $$d^{2}W/{d\xi }^2=\left(u^2{\xi }^m+\psi (\xi )\right)W,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $u$: large real or complex parameter, $W$: change of variable and $\psi (\xi )$: function

Referenced by:

§2.8(i), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E8

Encodings:

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Annotations for §2.8(i), §2.8 and Ch.2

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14: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

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30.9.1 $${\lambda }_n^m\left({\gamma }^2\right)\sim -{\gamma }^2+\gamma q+{\beta }_0+{\beta }_1{\gamma }^{-1}+{\beta }_2{\gamma }^{-2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and ${\beta }_n$: coeffients

A&S Ref:

21.7.6

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(i), §30.9 and Ch.30

… ►The cases of large $m$, and of large $m$ and large $|\gamma |$, are studied in Abramowitz (1949). …The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982).

15: 28.34 Methods of Computation

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

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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

16: 18.15 Asymptotic Approximations

… ►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 $0\le \beta /n\le c$. … ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … ►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: 30.11 Radial Spheroidal Wave Functions

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30.11.4 $$A_n^{\pm m}({\gamma }^2)=\sum _{2k\ge \mp m-n}{(-1)}^k{a}_{n,k}^{\pm m}({\gamma }^2)(\ne 0).$$

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Defines:

$A_n^m({\gamma }^2)$ (locally)

Symbols:

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

Referenced by:

§30.5, §30.6

Permalink:

http://dlmf.nist.gov/30.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(i), §30.11 and Ch.30

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§30.11(iii) Asymptotic Behavior

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30.11.6

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E6

Encodings:

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See also:

Annotations for §30.11(iii), §30.11 and Ch.30

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30.11.7 $$𝒲\left\{S_n^{m(1)}(z,\gamma ),S_n^{m(2)}(z,\gamma )\right\}=\frac{1}{\gamma (z^2-1)}.$$

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Symbols:

$𝒲$: Wronskian, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iv), §30.11 and Ch.30

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30.11.8 $$S_n^{m(1)}(z,\gamma )=K_n^m(\gamma ){\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$$

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Symbols:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.1 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E8

Encodings:

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See also:

Annotations for §30.11(v), §30.11 and Ch.30

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18: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

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§8.20(ii) Large $p$

… ►so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. …

19: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►In the following formulas for the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $u_k$, $v_k$ are the constants defined in §9.7(i), and $U_k(p)$, $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ► ►

§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

20: 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 ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. … ►

30.16.7 $$\sum _{j=1}^d{e}_{j,d}^2\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}=\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$$

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Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree and $d$: dimension

Permalink:

http://dlmf.nist.gov/30.16.E7

Encodings:

pMML, png, TeX

See also:

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

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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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