oscillatory%20integrals

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1: Bibliography K

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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

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§25.18(i).

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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Document, Code, ZentralBlatt

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§36.15(iii).

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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

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§32.11(iv).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

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§18.26(v), §18.26(v).

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2: Bibliography I

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

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§10.74(vii).

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An integral which occurs in statistics. Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.

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Document, ZentralBlatt

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§35.3(i), §35.3(ii).

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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MathReview (D. H. Lehmer), ZentralBlatt

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§24.12(i).

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Proceedings of ENuMath, Santiago de Compostela (2005)

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ISBN 3-540-34287-7, MathReview, ZentralBlatt

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§3.5(vii).

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

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§18.28(vii).

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3: Bibliography C

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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

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§26.8(vii).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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

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§13.29(v), §15.19(v).

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A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

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§36.15(iii), §36.8.

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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Document

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5th item.

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Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§19.13(i).

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4: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

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MathReview, ZentralBlatt

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§25.10(i), §27.11, §27.2(i).

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

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MathReview, ZentralBlatt

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§25.10(i), §27.11, §27.2(i).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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MathReview (Alan Weinstein), ZentralBlatt

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§36.12(iii).

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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5: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►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. … ►The bottom and top of the steps at the $x_i$ are lower and upper bounds to ${\int }_a^{x_i}d\mu (x)$ as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … ►The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$. …

6: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

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§31.10(i).

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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FullText, MathReview, ZentralBlatt

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§19.12, §19.5.

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Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).

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English translation in Functional Anal. Appl. 18 (1976), pp. 175-196.

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ISSN 0374-1990, Document, MathReview (Helmut Hamm), ZentralBlatt

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§36.6.

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

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§29.8, §29.8.

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

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§30.16(i), §30.16(ii), §30.16(ii).

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7: 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

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§22.19(i), §22.20(vi).

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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

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§28.8(iv).

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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt

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§2.5(ii).

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Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.

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ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt

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§2.9(iii).

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

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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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