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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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2: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. … ►Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).

3: 3.9 Acceleration of Convergence

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§3.9(i) Sequence Transformations

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§3.9(ii) Euler’s Transformation of Series

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§3.9(iv) Shanks’ Transformation

►Shanks’ transformation is a generalization of Aitken’s ${\mathrm{\Delta }}^2$-process. … ►

§3.9(vi) Applications and Further Transformations

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4: Bibliography

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

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Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

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Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§17.16, Ch.17.

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

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5: 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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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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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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External Links:

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

Referenced by:

§28.8(iv).

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Perturbational results for diffraction of water-waves by nearly-vertical barriers. IMA J. Appl. Math. 34 (1), pp. 99–117.

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External Links:

ISSN 0272-4960, Document, MathReview, ZentralBlatt

Referenced by:

§11.12.

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6: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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§22.7(iii) Generalized Landen Transformations

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7: Bibliography K

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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:

ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

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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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External Links:

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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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

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External Links:

ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt

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§26.1, §26.1, Notations, Notations.

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

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8: Bibliography L

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Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.

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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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External Links:

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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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

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4th item.

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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9: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►Rational solutions of ${\text{P}}_{\text{II}}$ exist for $\alpha =n(\in \mathbb{Z})$ and are generated using the seed solution $w(z;0)=0$ and the Bäcklund transformations (32.7.1) and (32.7.2). … ►

32.8.3 $$w(z;3)=\frac{3z^2}{z^3+4}-\frac{6z^2(z^3+10)}{z^6+20z^3-80},$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E3

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See also:

Annotations for §32.8(ii), §32.8 and Ch.32

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32.8.4 $$w(z;4)=-\frac{1}{z}+\frac{6z^2(z^3+10)}{z^6+20z^3-80}-\frac{9z^5(z^3+40)}{z^9+60z^6+11200}.$$

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Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E4

Encodings:

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See also:

Annotations for §32.8(ii), §32.8 and Ch.32

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$$Q_3(z)=z^6+20z^3-80,$$

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10: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

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