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11: 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
12: 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). …
13: 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 α , β , γ , and δ , the general solutions of P I P VI  are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations P II P VI  have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …
14: Bibliography F
  • FDLIBM (free C library)
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • 15: 25.12 Polylogarithms
    §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 …
    16: Bibliography S
  • L. Z. Salchev and V. B. Popov (1976) A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.
  • A. Salem (2013) Some properties and expansions associated with the q -digamma function. Quaest. Math. 36 (1), pp. 67–77.
  • I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
  • D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
  • R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.
  • 17: Bibliography R
  • S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • M. Robnik (1980) An extremum property of the n -dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.
  • K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.
  • 18: Bibliography H
  • D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
  • G. J. Heckman (1991) An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.
  • P. W. Hemker, T. H. Koornwinder, and N. M. Temme (1993) Wavelets: mathematical preliminaries. In Wavelets: an elementary treatment of theory and applications, Ser. Approx. Decompos., Vol. 1, pp. 13–26.
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
  • J. Humblet (1984) Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.
  • 19: Bibliography M
  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
  • S. M. Markov (1981) On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.
  • X. Merrheim (1994) The computation of elementary functions in radix 2 p . Computing 53 (3-4), pp. 219–232.
  • S. C. Milne (1985b) An elementary proof of the Macdonald identities for A l ( 1 ) . Adv. in Math. 57 (1), pp. 34–70.
  • J. Muller (1997) Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.
  • 20: 19.16 Definitions
    Just as the elementary function R C ( x , y ) 19.2(iv)) is the degenerate case … The R -function is often used to make a unified statement of a property of several elliptic integrals. …