fractional%20integrals

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1: Bibliography K

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

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2: 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. … ►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 $20$. … ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … ►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: …

3: Bibliography R

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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.

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

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

Referenced by:

§16.4(iii), §16.4(iii).

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

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4: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

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

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7th item.

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

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Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

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3rd item.

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Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.

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Notes:

Double-precision Fortran, maximum accuracy 16D

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Document, Code

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10th item.

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5: 20.11 Generalizations and Analogs

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …

6: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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Document

Referenced by:

§33.22(iv).

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Numerical evaluation of continued fractions. SIAM Rev. 6 (4), pp. 383–421.

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Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§3.10(iii), §3.10(ii).

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Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12 (4), pp. 519–526.

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ISSN 0272-4979, MathReview, ZentralBlatt

Referenced by:

§5.21, §5.24(ii).

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7: Bibliography C

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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

Referenced by:

§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

Referenced by:

§10.77(ix).

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.

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ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt

Referenced by:

§22.20(vii), §22.21.

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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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8: Bibliography L

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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

Referenced by:

§4.13, 2nd item, §4.48(iv).

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

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Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:

§3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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Continued Fractions with Applications. Studies in Computational Mathematics, North-Holland Publishing Co., Amsterdam.

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ISBN 0-444-89265-6, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iv), §1.12(v), §15.7, §18.13, §3.10(ii), §3.10(iii), §3.11(iv), §4.25, §4.39, §4.9(i), §4.9(ii), §5.10, §6.9, §7.18(v), §7.9.

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Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

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Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

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9: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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ISSN 0020-2932, Document, MathReview, ZentralBlatt

Referenced by:

§2.6(iii).

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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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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10: Bibliography W

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

Referenced by:

§29.19(ii).

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

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Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.22(vi), §10.23(iv), §10.43(vi).

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A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

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