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11: William P. Reinhardt

… ►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …

12: Bibliography

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

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A general mode theory for the elliptic disk microstrip antenna. IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.

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External Links:

Document

Referenced by:

5th item.

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Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:

Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.

External Links:

Document, MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

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Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

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External Links:

ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§17.13, §17.13.

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13: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.1 $$\arcsin z=z+\frac{1}{2}\frac{z^3}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{z^5}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^7}{7}+\mathrm{\cdots },$$ $|z|\le 1$.

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Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.40 (where the constraint is .)

Permalink:

http://dlmf.nist.gov/4.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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14: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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External Links:

ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

External Links:

MathReview

Referenced by:

§19.39(iii).

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

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Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:

Double-precision Fortran-90 code.

External Links:

Document, Code

Referenced by:

§33.23(vi), 8th item.

Encodings:

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15: Bibliography C

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Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

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Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

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External Links:

ISSN 0018-9480, FullText

Referenced by:

§10.21(x).

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Catastrophes and molecular collisions. Molecular Phys. 31 (1), pp. 33–55.

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External Links:

Document, MathReview (V. M. Zakaljukin)

Referenced by:

§36.14(iii).

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Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

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External Links:

ISSN 1076-2787, Document, MathReview

Referenced by:

§19.6(i).

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$S$-matrix and $R$-matrix determination of the low-energy ${}_{}{}^5\mathrm{He}$ and ${}_{}{}^5\mathrm{Li}$ resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

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External Links:

Document

Referenced by:

2nd item.

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16: 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.

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Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.

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External Links:

ISSN 0001-5962, Document, MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii).

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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).

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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).

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Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:

MathReview (F. Gotze), ZentralBlatt

Referenced by:

§13.31(iii).

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17: 3.9 Acceleration of Convergence

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Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
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2	0.82111 11111 11	0.82243 44785 14	0.82246 61821 45	0.82246 70102 48	0.82246 70327 79
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6	0.82239 19390 77	0.82246 61352 37	0.82246 70190 76	0.82246 70331 54	0.82246 70334 18
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… ►For examples and other transformations for convergent sequences and series, see Wimp (1981, pp. 156–199), Brezinski and Redivo Zaglia (1991, pp. 55–72), and Sidi (2003, Chapters 6, 12–13, 15–16, 19–24, and pp. 483–492). …

18: 10.60 Sums

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). … ►See also Watson (1944, Chapters 11 and 16).

19: Bibliography K

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Über die Wurzeln einiger Zylinderfunktionen und gewisser aus ihnen gebildeter Gleichungen. Zeitschrift für Mathematik und Physik 54, pp. 55–86 (German).

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External Links:

ZentralBlatt

Referenced by:

§10.21(x).

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

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Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 26–36

External Links:

ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 4th item, 5th item.

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

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Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§25.21(v).

Encodings:

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20: Bibliography D

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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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External Links:

ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

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Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

External Links:

ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

External Links:

ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141 (1), pp. 55–147.

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External Links:

ISSN 0020-9910, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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External Links:

ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

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