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21: Bibliography L

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:

ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

Encodings:

BibTeX

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Note sur la fonction $𝔎(w,x,s)={\sum }_{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{{(w+k)}^s}$ . Acta Math. 11 (1-4), pp. 19–24 (French).

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§25.14(i), Notations K.

Encodings:

BibTeX

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Global asymptotics of the Hahn polynomials. Anal. Appl. (Singap.) 11 (3), pp. 1350018, 47.

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External Links:

ISSN 0219-5305, Document, FullText, MathReview (Yamilet Quintana), ZentralBlatt

Referenced by:

§18.24, §18.24.

Encodings:

BibTeX

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Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iv).

Encodings:

BibTeX

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On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

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Notes:

In Russian. English translation: Differential Equations 3(11), pp. 994–999

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).

Encodings:

BibTeX

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22: 26.2 Basic Definitions

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Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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6	11	23	1255	40	37338
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9	30	26	2436	43	63261
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11	56	28	3718	45	89134
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13	101	30	5604	47	1 24754
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23: 30.7 Graphics

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Figure 30.7.4: Eigenvalues ${\lambda }_n^{10}\left({\gamma }^2\right)$, $n=10,11,12,13$, $-50\le {\gamma }^2\le 150$. Magnify

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Figure 30.7.5: ${\mathrm{𝖯𝗌}}_n^0(x,4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.6: ${\mathrm{𝖯𝗌}}_n^0(x,-4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.7: ${\mathrm{𝖯𝗌}}_n^1(x,30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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Figure 30.7.8: ${\mathrm{𝖯𝗌}}_n^1(x,-30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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24: Bibliography S

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

Referenced by:

8th item.

Encodings:

BibTeX

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On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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External Links:

ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§29.17(i), §29.8.

Encodings:

BibTeX

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

Encodings:

BibTeX

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A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.

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External Links:

ZentralBlatt

Referenced by:

§28.8(ii).

Encodings:

BibTeX

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25: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor $A_{jk}$ of $a_{jk}$ is … ►For real-valued $a_{jk}$, … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►If ${𝐷}_n[a_{j,k}]$ tends to a limit $L$ as $n\to \mathrm{\infty }$, then we say that the infinite determinant ${𝐷}_{\mathrm{\infty }}[a_{j,k}]$ converges and ${𝐷}_{\mathrm{\infty }}[a_{j,k}]=L$. … ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. …

26: Bibliography O

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Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{\mathrm{II}}$ and $P_{\mathrm{IV}}$ . Math. Ann. 275 (2), pp. 221–255.

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External Links:

ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).

Encodings:

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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External Links:

ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

Encodings:

BibTeX

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Studies on the Painlevé equations. IV. Third Painlevé equation $P_{\mathrm{III}}$ . Funkcial. Ekvac. 30 (2-3), pp. 305–332.

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External Links:

ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).

Encodings:

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Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

BibTeX

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Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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27: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. … ►These are given by the following equations in which $a_1,a_2,\mathrm{\dots },a_n$ are nonnegative integers such that …$M_1$ is the multinominal coefficient (26.4.2): …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. … ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …

28: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

29: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►When any one of $j_1,j_2,j_3$ is equal to $0,\frac{1}{2}$, or $1$, the $\mathit{3}j$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_1,j_2,j_3$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). … ►Even permutations of columns of a $\mathit{3}j$ symbol leave it unchanged; odd permutations of columns produce a phase factor ${(-1)}^{j_1+j_2+j_3}$, for example, … ►See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there. … ►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. …

30: 28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►

${\sigma }_m\sim 1+{\displaystyle \frac{s}{2^{3}h}}+{\displaystyle \frac{4s^2+3}{2^7{h}^2}}+{\displaystyle \frac{19s^3+59s}{2^{11}{h}^3}}+\mathrm{\cdots },$

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28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

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Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

… ►The approximations are expressed in terms of Whittaker functions $W_{\kappa ,\mu }\left(z\right)$ and $M_{\kappa ,\mu }\left(z\right)$ with $\mu =\frac{1}{4}$; compare §2.8(vi). … ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …