Pringsheim theorem for continued fractions

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21: 8.9 Continued Fractions

§8.9 Continued Fractions

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22: 4.9 Continued Fractions

§4.9 Continued Fractions

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§4.9(i) Logarithms

… ►For other continued fractions involving logarithms see Lorentzen and Waadeland (1992, pp. 566–568). … ►

§4.9(ii) Exponentials

… ►For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). …

23: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

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§24.6.

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

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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§5.10.

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

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Notes:

Fortran code.

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ISSN 0098-3500, Document

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§10.77(ix).

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

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Referenced by:

Profile Annie A. M. Cuyt.

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Handbook of Continued Fractions for Special Functions. Springer, New York.

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External Links:

ISBN 978-1-4020-6948-2, MathReview Entry, ZentralBlatt

Referenced by:

§10.10, §10.33, §10.55, §10.74(v), §12.6, §13.17, §13.5, §15.7, §16.4(iv), §17.6(vi), §17.7(ii), §18.13, §3.10(ii), §3.10(iii), §4.25, §4.39, §4.9(i), §4.9(ii), §4.9(iii), §5.10, §5.15, §6.9, §7.18(v), §7.22(i), §7.9, §8.17(v), §8.19(vii), §8.9.

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24: Wadim Zudilin

… ►Zudilin is author or coauthor of numerous publications including the book Neverending Fractions, An Introduction to Continued Fractions published by Cambridge University Press in 2014. …

25: Bibliography M

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

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§17.15.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

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

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

Referenced by:

§5.10.

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26: 8.25 Methods of Computation

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§8.25(iv) Continued Fractions

►The computation of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). …

27: 33.8 Continued Fractions

§33.8 Continued Fractions

… ►The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho \ne 0$. … ►The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. …

28: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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29: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►The $p_n^{(0)}(x)$ are also referred to as the numerator polynomials, the $p_n(x)$ then being the denominator polynomials, in that the $n$-th approximant of the continued fraction, $z\in ℂ$, … ►

Markov’s Theorem

►The ratio $p_n^{(0)}(z)/p_n(z)$, as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. … ►See Ismail (2009, p. 46 ), where the $k$th corecursive polynomial is also related to an appropriate continued fraction, given here as its $n$th convergent, …

30: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …