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1: 11.14 Tables

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${𝐇}_n\left(x\right)$, ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, and $I_n\left(x\right)-{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.1)5$, $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Agrest et al. (1982) tabulates ${𝐇}_n\left(x\right)$ and ${\mathrm{e}}^{-x}{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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Zanovello (1975) tabulates ${𝐇}_n\left(x\right)$ for $n=-4(1)15$ and $x=0.5(.5)26$ to 8D or 9S.

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Zhang and Jin (1996) tabulates ${𝐇}_n\left(x\right)$ and ${𝐋}_n\left(x\right)$ for $n=-4(1)3$ and $x=0(1)20$ to 8D or 7S.

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Agrest et al. (1982) tabulates ${\int }_0^x{𝐇}_0\left(t\right)dt$ and ${\mathrm{e}}^{-x}{\int }_0^x{𝐋}_0\left(t\right)dt$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

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2: Charles W. Clark

… ► H. … H. … H. … ►Clark received the R&D 100 Award, Distinguished Presidential Rank Award of the U. …

3: 23.17 Elementary Properties

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$\eta \left(\mathrm{i}\right)={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)}{2{\pi }^{3/4}}},$

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$\eta \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)={\displaystyle \frac{3^{1/8}{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^{3/2}}{2\pi }}{\mathrm{e}}^{\pi \mathrm{i}/24}.$

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23.17.6 $$\eta \left(\tau \right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{{(6n+1)}^2/12}.$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

►In (23.17.5) for terms up to $q^{48}$ see Zuckerman (1939), and for terms up to $q^{100}$ see van Wijngaarden (1953). … ►

23.17.8 $$\eta \left(\tau \right)=q^{1/12}\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n}),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.19

Permalink:

http://dlmf.nist.gov/23.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

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4: 10.75 Tables

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Döring (1971) tabulates the first 100 values of $\nu$ $(>1)$ for which $J_{-\nu }^{\prime }\left(x\right)$ has the double zero $x=\nu$, 10D.

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Döring (1966) tabulates all zeros of $Y_0\left(z\right)$, $Y_1\left(z\right)$, $H_0^{(1)}\left(z\right)$, $H_1^{(1)}\left(z\right)$, that lie in the sector , $|\mathrm{ph}z|\le \pi$, to 10D. Some of the smaller zeros of $Y_n\left(z\right)$ and $H_n^{(1)}\left(z\right)$ for $n=2,3,4,5,15$ are also included.

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Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of $H_0^{(1)}\left(z\right)$ and $H_1^{(1)}\left(z\right)$, 8D.

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Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros $\nu$ of $Y_{\nu }^{\prime }\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)$ and $H_{\nu }^{(1)}{}_{}{}^{\prime }\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)$, 8D.

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The main tables in Abramowitz and Stegun (1964, Chapter 10) give ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)8$, $x=0(.1)10$, 5–8S; ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0,1,2$, $x=0(.1)5$, 4–9D; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S. (For the notation see §10.1 and §10.47(ii).)

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5: 33.20 Expansions for Small $|ϵ|$

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$h(0,\mathrm{\ell };r)=-{(2r)}^{1/2}{Y}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right)$ , $r>0$,

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33.20.7 $$h(ϵ,\mathrm{\ell };r)\sim -A(ϵ,\mathrm{\ell })\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{𝖧}_k(\mathrm{\ell };r),$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and ${𝖧}_k(\mathrm{\ell };r)$: coefficient

Permalink:

http://dlmf.nist.gov/33.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(iii), §33.20 and Ch.33

►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and ►

33.20.8 $${𝖧}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{Y}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $C_{k,p}$: coefficients and ${𝖧}_k(\mathrm{\ell };r)$: coefficient

Permalink:

http://dlmf.nist.gov/33.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(iii), §33.20 and Ch.33

… ►For a comprehensive collection of asymptotic expansions that cover $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $ϵ\to 0\pm$ and are uniform in $r$, including unbounded values, see Curtis (1964a, §7). …

6: 28.22 Connection Formulas

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28.22.7 $$g_{o,2m+1}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)}{h{B}_1^{2m+1}(h^2)},$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

… ►where $A_n^m(h^2)$, $B_n^m(h^2)$ are as in §28.4(i), and $C_m(h^2)$, $S_m(h^2)$ are as in §28.5(i). … ►

$${\mathrm{fe}}_m^{\prime }(0,h^2)=\frac{1}{2}\pi C_m(h^2){\left(g_{e,m}(h)\right)}^2{\mathrm{ce}}_m(0,h^2),$$

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$${\mathrm{ge}}_m(0,h^2)=\frac{1}{2}\pi S_m(h^2){\left(g_{o,m}(h)\right)}^2{\mathrm{se}}_m^{\prime }(0,h^2).$$

… ►Here ${\mathrm{me}}_{\nu }(0,h^2)$ $(\ne 0)$ is given by (28.14.1) with $z=0$, and ${\mathrm{M}}_{\nu }^{(1)}(0,h)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu }(h^2)|=\max (|c_{2\mathrm{\ell }}^{\nu }(h^2)|)$, where the maximum is taken over all integers $\mathrm{\ell }$. …

7: 10.73 Physical Applications

… ►See Jackson (1999, Chapter 3, §§3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, §§100–102; Chapter VIII, §§186, 191–193; Chapter X, §§303, 304), Happel and Brenner (1973, Chapter 3, §3.3; Chapter 7, §7.3), Korenev (2002, Chapter 4, §43), and Gray et al. (1922, Chapter XI). … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

8: 10.4 Connection Formulas

… ►Other solutions of (10.2.1) include $J_{-\nu }\left(z\right)$, $Y_{-\nu }\left(z\right)$, $H_{-\nu }^{(1)}\left(z\right)$, and $H_{-\nu }^{(2)}\left(z\right)$. … ►

$$H_{-n}^{(1)}\left(z\right)={(-1)}^n{H}_n^{(1)}\left(z\right),$$

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$$H_{-n}^{(2)}\left(z\right)={(-1)}^n{H}_n^{(2)}\left(z\right).$$

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$$J_{\nu }\left(z\right)=\frac{1}{2}\left(H_{\nu }^{(1)}\left(z\right)+H_{\nu }^{(2)}\left(z\right)\right),$$

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$$H_{-\nu }^{(1)}\left(z\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}{H}_{\nu }^{(1)}\left(z\right),$$

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9: Bibliography

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Ly$\alpha$ line formation in starbursting galaxies. I. Moderately thick, dustless, and static H i media. Astrophysical J. 554, pp. 604–614.

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External Links:

FullText

Referenced by:

§7.21.

Encodings:

BibTeX

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Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.

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External Links:

ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§32.10(ii), §32.8(i), §32.8(v).

Encodings:

BibTeX

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Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:

Translated from the Russian by H. H. McFaden

External Links:

ISBN 0-8218-4524-1, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

Encodings:

BibTeX

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Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs, Vol. 79, American Mathematical Society, Providence, RI.

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Notes:

Translated from the second Russian edition by H. H. McFaden

External Links:

ISBN 0-8218-4532-2, MathReview (Glenn Stevens), ZentralBlatt

Referenced by:

§22.18(ii), §22.18(iv).

Encodings:

BibTeX

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Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

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Notes:

Erratum: same journal 17 (1975), no. 7, p. 1361

External Links:

Document, MathReview (K. Veselic), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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10: 10.19 Asymptotic Expansions for Large Order

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10.19.2 $$Y_{\nu }\left(z\right)\sim -\mathrm{i}{H}_{\nu }^{(1)}\left(z\right)\sim \mathrm{i}{H}_{\nu }^{(2)}\left(z\right)\sim -\sqrt{\frac{2}{\pi \nu }}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{-\nu }.$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.3.1

Referenced by:

§10.41(i)

Permalink:

http://dlmf.nist.gov/10.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(i), §10.19 and Ch.10

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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$P_2(a)=-\frac{9}{100}{a}^5+\frac{3}{35}{a}^2,$

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10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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$R_2(a)=-\frac{9}{100}{a}^5+\frac{57}{70}{a}^2,$

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