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1: 22.6 Elementary Identities
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
2: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
§4.29(ii) Complex Arguments
The conformal mapping w = sinh z is obtainable from Figure 4.15.7 by rotating both the w -plane and the z -plane through an angle 1 2 π , compare (4.28.8). …
3: 16.24 Physical Applications
The 3 j symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as F 2 3 functions with unit argument. …These are balanced F 3 4 functions with unit argument. Lastly, special cases of the 9 j symbols are F 4 5 functions with unit argument. …
4: Bibliography B
  • L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1997) New tables of Bessel functions of complex argument. Comput. Math. Math. Phys. 37 (12), pp. 1480–1482.
  • 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.
  • R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.
  • W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
  • A. I. Burshtein and S. I. Temkin (1994) Spectroscopy of Molecular Rotation in Gases and Liquids. Cambridge University Press, Cambridge.
  • 5: Bibliography T
  • S. A. Teukolsky (1972) Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.
  • I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 159 (3), pp. 241–242.
  • I. Thompson (2013) Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.
  • W. J. Thompson (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
  • J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.
  • 6: 33.22 Particle Scattering and Atomic and Molecular Spectra
    The Coulomb functions given in this chapter are most commonly evaluated for real values of ρ , r , η , ϵ and nonnegative integer values of , but they may be continued analytically to complex arguments and order as indicated in §33.13. …
  • Eigenstates using complex-rotated coordinates r r e i θ , so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

  • 7: Bibliography K
  • M. Kodama (2008) Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
  • M. Kodama (2011) Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
  • P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.
  • K. S. Kölbig (1972c) Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.
  • S. Kowalevski (1889) Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).
  • 8: 22.19 Physical Applications
    22.19.2 sin ( 1 2 θ ( t ) ) = sin ( 1 2 α ) sn ( t + K , sin ( 1 2 α ) ) ,
    22.19.3 θ ( t ) = 2 am ( t E / 2 , 2 / E ) ,
    See accompanying text
    Figure 22.19.1: Jacobi’s amplitude function am ( x , k ) for 0 x 10 π and k = 0.5 , 0.9999 , 1.0001 , 2 . When k < 1 , am ( x , k ) increases monotonically indicating that the motion of the pendulum is unbounded in θ , corresponding to free rotation about the fulcrum; compare Figure 22.16.1. … Magnify
    The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …
    9: DLMF Project News
    error generating summary
    10: Bibliography P
  • B. V. Pal tsev (1999) On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.