integral%20equations%20and%20representations

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1: 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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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

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An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.

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ISSN 0377-0427, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§18.34(ii).

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An integral representation for the Lambert $W$ function.

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External Links:

2012.02480

Referenced by:

(4.13.16), §4.13.

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The Airy Integral, Giving Tables of Solutions of the Differential Equation $y^{\prime \prime }=xy$ . British Association for the Advancement of Science, Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.

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External Links:

MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

1st item, 1st item, §9.2(i), §9.2(ii), §9.2(v), §9.4, (9.6.2), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.8(i), §9.8(ii), §9.8(iv), §9.9(ii), Ch.9.

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2: Bibliography C

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On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

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Notes:

Includes Fortran code

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Document, ZentralBlatt

Referenced by:

§8.28(vi).

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

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On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

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ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt

Referenced by:

§14.11, §14.11.

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A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.

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ISSN 0026-8976, Document, MathReview (Peter Giblin)

Referenced by:

§36.10(i), §36.10(ii), §36.15(v), §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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3: Bibliography K

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Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.

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ISSN 1073-2772, FullText, MathReview, ZentralBlatt

Referenced by:

§31.10(ii).

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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

Referenced by:

1st item.

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Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

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Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.

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ISSN 0893-4983, MathReview, ZentralBlatt

Referenced by:

§32.8(v).

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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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4: Bibliography V

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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

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ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

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Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.27.

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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

Referenced by:

§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

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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5: 25.12 Polylogarithms

… ►The remainder of the equations in this subsection apply to principal branches. … ►The right-hand side is called Clausen’s integral. … ►

Integral Representation

… ►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms …

6: 19.36 Methods of Computation

… ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … ►

7: 25.11 Hurwitz Zeta Function

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§25.11(iii) Representations by the Euler–Maclaurin Formula

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§25.11(iv) Series Representations

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§25.11(vii) Integral Representations

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§25.11(viii) Further Integral Representations

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§25.11(x) Further Series Representations

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8: 14.30 Spherical and Spheroidal Harmonics

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Explicit Representation

… ►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). … ►As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): … ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations …

9: Bibliography G

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Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.

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ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.7(ii).

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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

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Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.18.

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

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

Referenced by:

3rd item, §10.77(ix).

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Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique. Ann. Sci. École Norm. Sup. (2) 10, pp. 3–142 (French).

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External Links:

ISSN 0012-9593, MathReview Entry, ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

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10: 36.5 Stokes Sets

… ►where $x_{\pm }$ are the two smallest positive roots of the equation … ►where $u$ satisfies the equation … ►Here $u$ is the root of the equation … ►where $u$ is the root of the equation … ►where $u$ is the positive root of the equation …