Sears? balanced 4?3 transformation

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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: 3.5 Quadrature

… ►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … ►

Example. Laplace Transform Inversion

… ►A special case is the rule for Hilbert transforms (§1.14(v)): … ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

3: Bibliography B

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Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.

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Referenced by:

§36.14(i).

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Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

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External Links:

ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§18.2(xii).

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The SLATEC Common Mathematical Library. In Sources and Development of Mathematical Software, W. R. Cowell (Ed.), pp. 302–320.

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External Links:

GAMS (package SLATEC)

Referenced by:

SLATEC (free Fortran library).

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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

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Elements of the Integral Calculus. 2nd edition, Ginn & Co., Boston.

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Notes:

Reprinted by G. E. Stechert, New York, 1926

Referenced by:

§19.11(i), Ch.19.

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4: Bibliography C

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A note on the asymptotic expansion of the Lerch’s transcendent. Integral Transforms Spec. Funct. 30 (10), pp. 844–855.

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ISSN 1065-2469, Document, Link, MathReview (Siddhi S. Pathak)

Referenced by:

§25.14(ii).

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A History of Mathematical Notations, Volume II. Open Court Publishing Company, Chicago.

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Notes:

Reprinted by Cosimo in 2007.

External Links:

ZentralBlatt

Referenced by:

§24.1.

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Expansions of $q$-Bernoulli numbers. Duke Math. J. 25 (2), pp. 355–364.

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External Links:

Document, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§17.3(iii).

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Eigenvalues of Density Matrices. Phys. Rev. 121, pp. 659–661.

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External Links:

Document, Link, ZentralBlatt

Referenced by:

Profile Bille C. Carlson.

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Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:

with Microfiche Supplement

External Links:

FullText, MathReview, ZentralBlatt

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3rd item.

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5: Bibliography N

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Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:

ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(25.11.44), (25.11.45), §25.11(xii), §25.11(xii), §8.15, Erratum (V1.2.1) for §25.11(xii), Erratum (V1.2.1) for Addition.

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Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.

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ISSN 0036-1410, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(10.18.17), (10.18.18), §10.18(iii), (9.8.22), (9.8.23), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv), Erratum (V1.1.8) for Section 10.18(iii).

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Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

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ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

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1st item.

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Eigenstates, coherent states, and uncertainty products for the Morse oscillator. Phys. Rev. A (3) 19 (2), pp. 438–444.

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ISSN 1050-2947, Document, Link, MathReview Entry

Referenced by:

(18.39.17), (18.39.18).

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Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:

ISSN 1364-5021, Document, ZentralBlatt

Referenced by:

§36.7(iv).

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6: Bibliography M

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The ster and stei functions. Phil. Mag. Series 7 21 (140), pp. 425–436.

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External Links:

Document, ZentralBlatt

Referenced by:

§11.8.

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The generalized integro-exponential function. Math. Comp. 44 (170), pp. 443–458.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§8.19(v), §8.19(xi).

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On the choice of standard solutions to Weber’s equation. Proc. Cambridge Philos. Soc. 48, pp. 428–435.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§12.1, §12.14(i), §12.2(i).

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

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7: 16.24 Physical Applications

… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6}j$ symbols. These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. …

8: 16.4 Argument Unity

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Pfaff–Saalschütz Balanced Sum

… ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ►Contiguous balanced series have parameters shifted by an integer but still balanced. … …

9: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

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Werke. Band II. pp. 436–447 (German).

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Notes:

Reprinted by Georg Olms Verlag, Hildesheim, 1973.

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§27.2(i).

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Applications of the classical umbral calculus. Algebra Universalis 49 (4), pp. 397–434.

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ISSN 0002-5240, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§24.4(viii).

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Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (43), pp. 43016, 16 pp..

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ISSN 1751-8113, Document, Link, MathReview (María-José Cantero)

Referenced by:

§18.36(vi).

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Mémoire sur les fonctions hypergéométriques d’ordre supérieur. Ann. Sci. École Norm. Sup. (2) 12, pp. 261–286, 395–430 (French).

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External Links:

ZentralBlatt

Referenced by:

§16.12.

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10: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

§17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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F. H. Jackson’s Transformations

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Sears’ Balanced ${}_4{}^{}\varphi _3^{}$ Transformations

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Sears–Carlitz Transformation

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Mixed-Base Heine-Type Transformations

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