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21: 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).

Encodings:

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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).

Encodings:

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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).

Encodings:

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22: Bibliography B

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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§8.7.

Encodings:

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New “coherent” states associated with non-compact groups. Comm. Math. Phys. 21 (1), pp. 41–55.

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External Links:

ISSN 0010-3616, MathReview (R. Mirman), ZentralBlatt

Referenced by:

§13.28(iii).

Encodings:

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

Encodings:

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

Encodings:

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Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

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23: 8.21 Generalized Sine and Cosine Integrals

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8.21.4 $$\mathrm{si}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\sin tdt,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{si}(a,z)$: generalized sine integral, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.8 (This function is called $S(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E4

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Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.5 $$\mathrm{ci}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\cos tdt,$$ ,

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{ci}(a,z)$: generalized cosine integral, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.7 (This function is called $C(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E5

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Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

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Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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24: 10.12 Generating Function and Associated Series

… ►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, …

25: 22.15 Inverse Functions

… ►Comprehensive treatments are given by Carlson (2005), Lawden (1989, pp. 52–55), Bowman (1953, Chapter IX), and Erdélyi et al. (1953b, pp. 296–301). …

26: Bibliography M

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On the computation of zeroes of $J_n(z)-i{J}_{n+1}(z)=0$ . Quart. Appl. Math. 55 (4), pp. 623–633.

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External Links:

ISSN 0033-569X, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

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Spheroidal eigenfunctions of the tidal equation. Phys. Rev. Lett. 73 (11), pp. 1557–1560.

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

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Zeros of the function $A{i}^{\prime }(z)-\sigma Ai(z)$ . Differential Equations 11, pp. 797–811.

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External Links:

ZentralBlatt

Referenced by:

§9.9(i).

Encodings:

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27: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

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See also:

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

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4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

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See also:

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

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4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

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Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

Encodings:

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See also:

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

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4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

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See also:

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

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4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

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See also:

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

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28: 24.13 Integrals

… ►For other integrals see Prudnikov et al. (1990, pp. 55–57).

29: 10 Bessel Functions

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30: 8.2 Definitions and Basic Properties

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8.2.11 $$\mathrm{\Gamma }(a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mathrm{\Gamma }\left(a\right)(1-z^a{\mathrm{e}}^{\pm \pi \mathrm{i}a}{\gamma }^{\ast }(a,-z)).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E11

Encodings:

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See also:

Annotations for §8.2(ii), §8.2 and Ch.8

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