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1: 22.3 Graphics
See accompanying text
Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
See accompanying text
Figure 22.3.27: Density plot of | sn ( 10 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
See accompanying text
Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
See accompanying text
Figure 22.3.29: Density plot of | sn ( 30 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
2: 20.11 Generalizations and Analogs
This is Jacobi’s inversion problem of §20.9(ii). … Each provides an extension of Jacobi’s inversion problem. …
§20.11(iv) Theta Functions with Characteristics
For m = 1 , 2 , 3 , 4 , n = 1 , 2 , 3 , 4 , and m n , define twelve combined theta functions φ m , n ( z , q ) by …
3: 20.7 Identities
20.7.34 θ 1 ( z , q 2 ) θ 3 ( z , q 2 ) θ 1 ( z , i q ) = θ 2 ( z , q 2 ) θ 4 ( z , q 2 ) θ 2 ( z , i q ) = i 1 / 4 θ 2 ( 0 , q 2 ) θ 4 ( 0 , q 2 ) 2 .
4: 20.10 Integrals
§20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
Here ζ ( s ) again denotes the Riemann zeta function25.2). …
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …
5: Bibliography K
  • 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. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.
  • A. Khare and U. Sukhatme (2004) Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.
  • T. H. Koornwinder (1984a) Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In Special Functions: Group Theoretical Aspects and Applications, pp. 1–85.
  • 6: Bibliography B
  • W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • 7: Bibliography F
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • C. L. Frenzen and R. Wong (1985b) A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 8: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • Y. Chow, L. Gatteschi, and R. Wong (1994) A Bernstein-type inequality for the Jacobi polynomial. Proc. Amer. Math. Soc. 121 (3), pp. 703–709.
  • H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 9: Bibliography I
  • J. Igusa (1972) Theta Functions. Springer-Verlag, New York.
  • E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.
  • M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and q -Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
  • 10: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.