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1: 32.8 Rational Solutions
P II P VI  possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … Special rational solutions of P III  are … Special rational solutions of P IV  are … Special rational solutions of P V  are … These are special cases of §32.10(vi). …
2: 25.12 Polylogarithms
Other notations and names for Li 2 ( z ) include S 2 ( z ) (Kölbig et al. (1970)), Spence function Sp ( z ) (’t Hooft and Veltman (1979)), and L 2 ( z ) (Maximon (2003)). …
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . Principal value. … Magnify 3D Help
For other values of z , Li s ( z ) is defined by analytic continuation. … The special case z = 1 is the Riemann zeta function: ζ ( s ) = Li s ( 1 ) . …
3: 19.36 Methods of Computation
Complex values of the variables are allowed, with some restrictions in the case of R J that are sufficient but not always necessary. … This method loses significant figures in ρ if α 2 and k 2 are nearly equal unless they are given exact values—as they can be for tables. … Also, see Todd (1975) for a special case of K ( k ) . 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). … These special theorems are also useful for checking computer codes. …
4: Bibliography K
  • E. G. Kalnins, W. Miller, G. F. Torres del Castillo, and G. C. Williams (2000) Special Functions and Perturbations of Black Holes. In Special Functions (Hong Kong, 1999), pp. 140–151.
  • N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • 5: Bibliography I
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • Inverse Symbolic Calculator (website) Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.
  • M. E. H. Ismail and E. Koelink (Eds.) (2005) Theory and Applications of Special Functions. Developments in Mathematics, Vol. 13, Springer, New York.
  • M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions, q -Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
  • 6: Bibliography L
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • N. N. Lebedev (1965) Special Functions and Their Applications. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • D. W. Lozier and F. W. J. Olver (1994) Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.
  • Y. L. Luke (1969a) The Special Functions and their Approximations, Vol. 1. Academic Press, New York.
  • 7: Bibliography N
  • A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Németh (1992) Mathematical Approximation of Special Functions. Nova Science Publishers Inc., Commack, NY.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 8: 14.30 Spherical and Spheroidal Harmonics
    Special Values
    14.30.4 Y l , m ( 0 , ϕ ) = { ( 2 l + 1 4 π ) 1 / 2 , m = 0 , 0 , | m | = 1 , 2 , 3 , ,
    The special class of spherical harmonics Y l , m ( θ , ϕ ) , defined by (14.30.1), appear in many physical applications. …has solutions W ( ρ , θ , ϕ ) = ρ l Y l , m ( θ , ϕ ) , which are everywhere one-valued and continuous. …
    9: 3.8 Nonlinear Equations
    From this graph we estimate an initial value x 0 = 4.65 . … Table 3.8.3 gives the successive values of s and t . … For fixed-point methods for computing zeros of special functions, see Segura (2002), Gil and Segura (2003), and Gil et al. (2007a, Chapter 7). … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . …
    10: Bibliography G
  • F. G. Garvan and M. E. H. Ismail (Eds.) (2001) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.
  • W. Gautschi (1975) Computational Methods in Special Functions – A Survey. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.
  • W. Gautschi (1997b) The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.