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1: Bibliography G

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Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt

Referenced by:

2nd item.

Encodings:

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A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.25(iv), §8.25(v).

Encodings:

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Some integrals involving three modified Bessel functions. I. J. Math. Phys. 27 (3), pp. 682–687.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89

External Links:

Document, MathReview (A. Korányi), ZentralBlatt

Referenced by:

§35.3(i).

Encodings:

BibTeX

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The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:

In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(v).

Encodings:

BibTeX

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2: Bibliography I

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IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std ISO/IEC/IEEE 60559, Revision of IEEE Std 754-1985

External Links:

ISBN 978-2-88912-566-1, Document

Referenced by:

Figure 3.1.1, Figure 3.1.1, §3.1(i), §3.1(i), Erratum (V1.0.25) for Section 3.1.

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

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Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

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External Links:

ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.26(iv).

Encodings:

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

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3: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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External Links:

ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.58.

Encodings:

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.12, §8.25(iii).

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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.

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External Links:

ISSN 0021-9045, Document, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§24.11.

Encodings:

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4: Bibliography S

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Zur Berechnung der Eigenwerte der Sphäroiddifferentialgleichung. Numer. Math. 4, pp. 310–312 (German).

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External Links:

ISSN 0029-599X, Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§30.16(i).

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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On parabolic cylinder functions. J. Inst. Math. Appl. 4 (1), pp. 106–112.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§12.16.

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Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.

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External Links:

Document

Referenced by:

§8.24(i).

Encodings:

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5: 20.7 Identities

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012). … ►See also Carlson (2011, §§1 and 4). … ►

§20.7(vii) Derivatives of Ratios of Theta Functions

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20.7.25 $$\frac{d}{dz}\left(\frac{{\theta }_2\left(z\right|\tau )}{{\theta }_4\left(z\right|\tau )}\right)=-\frac{{\theta }_3^2\left(0\right|\tau ){\theta }_1\left(z\right|\tau ){\theta }_3\left(z\right|\tau )}{{\theta }_4^2\left(z\right|\tau )}.$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vii), §20.7 and Ch.20

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6: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►

§12.14(vii) Relations to Other Functions

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12.14.13 $$W(0,\pm x)=2^{-\frac{5}{4}}\sqrt{\pi x}\left(J_{-\frac{1}{4}}\left(\frac{1}{4}{x}^2\right)\mp J_{\frac{1}{4}}\left(\frac{1}{4}{x}^2\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $W(a,x)$: parabolic cylinder function and $x$: real variable

A&S Ref:

19.25.4 (corrected)

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/12.14.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

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12.14.14 $$\frac{d}{dx}W(0,\pm x)=-2^{-\frac{9}{4}}x\sqrt{\pi x}\left(J_{\frac{3}{4}}\left(\frac{1}{4}{x}^2\right)\pm J_{-\frac{3}{4}}\left(\frac{1}{4}{x}^2\right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $W(a,x)$: parabolic cylinder function and $x$: real variable

A&S Ref:

19.25.5

Permalink:

http://dlmf.nist.gov/12.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

… ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). …

7: 30.16 Methods of Computation

… ►The eigenvalues of $𝐀$ can be computed by methods indicated in §§3.2(vi), 3.2(vii). … ►For $m=2$, $n=4$, ${\gamma }^2=10$, … ►

${\alpha }_{2,4}=\mathrm{13.97907\hspace{0.33em}459},$

… ►which yields ${\lambda }_4^2\left(10\right)=\mathrm{13.97907\hspace{0.33em}345}$. … ►The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11). …

8: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

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On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

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Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27 (4), pp. 399–411.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.10(ii), §33.11, §33.12(i), §33.9(ii).

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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

Encodings:

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9: 18.13 Continued Fractions

… ►The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii). … ►

18.13.3 $$\frac{a_1}{x+}\frac{-\frac{1}{2}}{\frac{3}{2}x+}\frac{-\frac{2}{3}}{\frac{5}{3}x+}\frac{-\frac{3}{4}}{\frac{7}{4}x+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.4 $$\frac{a_1}{1-x+}\frac{-\frac{1}{2}}{\frac{1}{2}(3-x)+}\frac{-\frac{2}{3}}{\frac{1}{3}(5-x)+}\frac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.5 $$\frac{1}{2x+}\frac{-2}{2x+}\frac{-4}{2x+}\frac{-6}{2x+}\mathrm{\cdots }.$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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10: Bibliography B

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A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

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A new method for investigating Euler sums. Ramanujan J. 4 (4), pp. 397–419.

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External Links:

ISSN 1382-4090, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.6.16), (25.6.17), (25.6.18), (25.6.19), (25.6.20), §25.6(iii), §25.6(iii), §25.6.

Encodings:

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Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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External Links:

Document, MathReview (Bernard Harris), ZentralBlatt

Referenced by:

Ch.2.

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A uniform asymptotic formula for orthogonal polynomials associated with $\exp (-x^4)$ . J. Approx. Theory 98, pp. 146–166.

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External Links:

ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.32.

Encodings:

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Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 46(1991), no. 4, pp. 1–45

External Links:

ISSN 0042-1316, MathReview (P. E. Markov), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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