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1: 18.40 Methods of Computation

… ►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

… ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …

2: Bibliography W

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The analyticity of Jacobian functions with respect to the parameter $k$ . Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§22.17(ii), §22.2.

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Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

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Notes:

Reprinted by Chelsea Publishing, New York, 1973 and AMS Chelsea Publishing, Providence, RI, 2000.

External Links:

ISBN 0-8218-2106-7, MathReview (R. C. Buck), ZentralBlatt

Referenced by:

§3.10(iii), §4.25, §4.9(i), §4.9(ii), Ch.4, §5.10.

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-521-35883-3, MathReview, ZentralBlatt

Referenced by:

§22.19(i), §22.19(iv), §22.19(v), §23.21(i).

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3: Bibliography M

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The Theory of Analytic Functions: A Brief Course. “Mir”, Moscow.

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Notes:

Translated from the Russian by Eugene Yankovsky

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.10(iii), §1.10(vi), §1.9(i), §1.9(iv), §1.9(vi).

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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

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External Links:

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

Referenced by:

§5.10.

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

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External Links:

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

Referenced by:

§5.10.

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4: Bibliography R

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Topics in Analytic Number Theory. Springer-Verlag, New York.

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Notes:

Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169

External Links:

MathReview (H. Halberstam), ZentralBlatt

Referenced by:

§1.8(iv), §20.7(viii), §20.7(viii), Ch.24.

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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