Van Vleck theorem for continued fractions

SearchAdvancedHelp

(0.002 seconds)

11—20 of 266 matching pages

11: 13.17 Continued Fractions

§13.17 Continued Fractions

… ►

13.17.1 $$\frac{\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)}{M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.17 and Ch.13

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

12: Bibliography V

… ►

Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§30.16(ii), §30.18(iii).

Encodings:

BibTeX

… ►

Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

ⓘ

Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§28.36(iii).

Encodings:

BibTeX

►

Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

ⓘ

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

Encodings:

BibTeX

►

Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.

ⓘ

External Links:

MathReview (P. M. Morse)

Referenced by:

§9.16.

Encodings:

BibTeX

►

Multiple Light Scattering. Vol. 1, Academic Press, New York.

ⓘ

Referenced by:

§6.17.

Encodings:

BibTeX

…

13: 12.6 Continued Fraction

§12.6 Continued Fraction

►For a continued-fraction expansion of the ratio $U(a,x)/U(a-1,x)$ see Cuyt et al. (2008, pp. 340–341).

14: Bibliography M

… ►

The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

ⓘ

External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

… ►

Handbook of Applied Cryptography. CRC Press, Boca Raton, FL.

ⓘ

Notes:

With a foreword by Ronald L. Rivest

External Links:

ISBN 0-8493-8523-7, MathReview (Yvo Desmedt), ZentralBlatt

Referenced by:

§27.16.

Encodings:

BibTeX

… ►

Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

Encodings:

BibTeX

… ►

A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

ⓘ

External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

… ►

A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

ⓘ

External Links:

ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

…

15: Bibliography D

… ►

Integralen voor de $\zeta$-functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).

ⓘ

Notes:

In Dutch.

External Links:

ZentralBlatt

Referenced by:

(25.5.15), (25.5.16), (25.5.17), (25.5.18), (25.5.19), §25.5(ii).

Encodings:

BibTeX

… ►

Computing Riemann theta functions. Math. Comp. 73 (247), pp. 1417–1442.

ⓘ

Notes:

Two of the authors (Deconinck and van Hoeij) have developed Maple code for the algorithms described in this paper.

External Links:

ISSN 0025-5718, Document, Code, MathReview, ZentralBlatt

Referenced by:

§21.10(i), 1st item, §21.2(i), §21.4, M. Heil (1995).

Encodings:

BibTeX

… ►

Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

ⓘ

External Links:

ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

ⓘ

External Links:

ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

Encodings:

BibTeX

… ►

Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

Encodings:

BibTeX

…

16: Bibliography C

… ►

On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

… ►

Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

ⓘ

Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

… ►

Painlevé Equations—Nonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.

ⓘ

External Links:

ISBN 3-540-31062-2, MathReview (Marta Mazzocco), ZentralBlatt

Referenced by:

Ch.32.

Encodings:

BibTeX

… ►

Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:

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

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

… ►

Handbook of Continued Fractions for Special Functions. Kluwer Academic Publishers Group, Dordrecht.

ⓘ

Referenced by:

Profile Annie A. M. Cuyt.

Encodings:

BibTeX

…

17: Bibliography G

… ►

A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

ⓘ

External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

… ►

WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.

ⓘ

External Links:

FullText, MathReview (Ioannis K. Purnaras), ZentralBlatt

Referenced by:

§18.15(v), §18.15(v).

Encodings:

BibTeX

… ►

Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

ⓘ

External Links:

ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt

Referenced by:

§3.2(i), §3.2(i), §3.2(ii), §3.2(vii).

Encodings:

BibTeX

… ►

Table of ${\int }_0^{\mathrm{\infty }}\frac{e^{-u^2}}{u+x}𝑑u$ . Quart. J. Mech. Appl. Math. 1 (1), pp. 319–326.

ⓘ

External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§7.22(ii).

Encodings:

BibTeX

… ►

Inductance Calculations. Van Nostrand, New York.

ⓘ

Notes:

Reprinted by Dover, New York, 1962 and 2004

External Links:

ZentralBlatt

Referenced by:

§19.34.

Encodings:

BibTeX

…

18: 30.16 Methods of Computation

… ►Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93). … ►For other methods see Van Buren and Boisvert (2002, 2004).

19: Bibliography J

… ►

Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

ⓘ

External Links:

MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

Encodings:

BibTeX

… ►

Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

ⓘ

External Links:

FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

Encodings:

BibTeX

… ►

Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

ⓘ

External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

Encodings:

BibTeX

►

Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

ⓘ

External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

…

20: Bibliography B

… ►

Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.

ⓘ

External Links:

MathReview (L. Van Hove), ZentralBlatt

Referenced by:

§18.39(ii), §18.39(ii).

Encodings:

BibTeX

… ►

Quantum Theory of Angular Momentum. A Collection of Reprints and Original Papers. Academic Press, New York.

ⓘ

External Links:

ISBN 0-12-096056-7, MathReview (R. Mirman)

Referenced by:

§34.12, §34.3(v), §34.5(v), §34.7(v).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.

ⓘ

External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§2.3(v).

Encodings:

BibTeX

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

… ►

Faster Primality Testing. In Advances in Cryptology—EUROCRYPT ’89 Proceedings, J.-J. Quisquater and J. Vandewalle (Eds.), Lecture Notes in Computer Science, Vol. 434, New York, pp. 652–656.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§27.18.

Encodings:

BibTeX

…