elementary properties

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11: 18.36 Miscellaneous Polynomials

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. See Gómez-Ullate et al. (2009) for an elementary introduction. …

12: 3.11 Approximation Techniques

… ►For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. … ►A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). … ►The property …

13: 19.15 Advantages of Symmetry

… ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

14: 32.2 Differential Equations

… ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … ►There are fifty equations with the Painlevé property. … ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …

15: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

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16: 25.12 Polylogarithms

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§25.12(i) Dilogarithms

… ►The cosine series in (25.12.7) has the elementary sum … ►For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989). … ►

§25.12(ii) Polylogarithms

… ►Further properties include …

17: Bibliography C

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An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:

Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.

External Links:

ZentralBlatt

Referenced by:

§19.12, Ch.19, A. Cayley (1961).

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An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:

Second edition of Cayley (1895), unabridged and corrected

External Links:

ZentralBlatt

Referenced by:

§19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).

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Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

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Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.

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External Links:

ISBN 0138220646, ZentralBlatt

Referenced by:

§4.45(i).

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Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.

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External Links:

ISSN 0098-3500, Document, GAMS (module 714; package TOMS), ZentralBlatt

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1st item.

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18: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

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Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.

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External Links:

ISSN 0025-5718, FullText, MathReview (Menachem Dishon), ZentralBlatt

Referenced by:

§4.45(i).

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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External Links:

ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

Referenced by:

§18.37(iii).

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19: Bibliography R

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

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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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External Links:

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

Referenced by:

§16.4(iii), §16.4(iii).

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An extremum property of the $n$-dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.

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ISSN 0305-4470, Document, MathReview

Referenced by:

§5.19(iii).

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Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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External Links:

ISBN 0-201-87073-8, MathReview (Norman J. Richert)

Referenced by:

§27.17.

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20: Bibliography H

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

Referenced by:

§1.2(vi).

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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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An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

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External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

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Wavelets: mathematical preliminaries. In Wavelets: an elementary treatment of theory and applications, Ser. Approx. Decompos., Vol. 1, pp. 13–26.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Tom H. Koornwinder.

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Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.

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External Links:

ISSN 0003-4916, Document, MathReview

Referenced by:

§33.10(ii), §33.13, §33.22(vii).

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