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11: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

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12: 30.3 Eigenvalues

… ►If $p=0$ or $p=1$, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if $p>1$ its last denominator is ${\beta }_0-\lambda$ or ${\beta }_1-\lambda$. …

13: DLMF Project News

error generating summary

14: 17.4 Basic Hypergeometric Functions

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17.4.6 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n},$$

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Defines:

${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.4.6)

Permalink:

http://dlmf.nist.gov/17.4.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.3):

The explicit usage of ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

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15: 23.2 Definitions and Periodic Properties

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23.2.4 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{{(z-w)}^2}-\frac{1}{w^2}\right),$$

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Defines:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function

Symbols:

$\mathrm{\wp }(z;g_2,g_3)$: Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass $\mathrm{\wp }$-function

Referenced by:

§23.9, Erratum (V1.0.5) for Equation (23.2.4)

Permalink:

http://dlmf.nist.gov/23.2.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the denominator ${(z-w)}^2$ was given incorrectly as $(z-w^2)$.

Reported 2012-02-16 by James D. Walker

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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16: 18.2 General Orthogonal Polynomials

… ►Because of (18.2.36) the OP’s $p_n(x)$ are also called monic denominator polynomials and the OP’s $p_{n-1}^{(1)}(x)$, or, equivalently, the $p_n^{(0)}(x)$, are called the monic numerator polynomials. …

17: 29.3 Definitions and Basic Properties

… ►The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if $p=0$, and if $p>0$, then the last denominator is ${\beta }_0-H$. …

18: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function ${}_p{}^{}F_q^{}$ with matrix argument $𝐓\in 𝓢$, numerator parameters $a_1,\mathrm{\dots },a_p$, and denominator parameters $b_1,\mathrm{\dots },b_q$ is …

19: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. … ►

The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right)$$

Originally, the factor of 2 was missing from the denominator of the argument of the $\cot$ function.

Reported by Blagoje Oblak on 2019-05-27

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k}$$

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported by Ian Thompson on 2018-05-17

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23.2.4 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{{(z-w)}^2}-\frac{1}{w^2}\right)$$

Originally the denominator ${(z-w)}^2$ was given incorrectly as $(z-w^2)$.

Reported 2012-02-16 by James D. Walker.

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20: 1.2 Elementary Algebra

… ►To find the polynomials $f_j(x)$, $j=1,2,\mathrm{\dots },n$, multiply both sides by the denominator of the left-hand side and equate coefficients. …