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1: 22.6 Elementary Identities
§22.6(iii) Half Argument
2: Bibliography F
  • T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.
  • 3: 10.77 Software
    §10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)
    §10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions
    4: 20.2 Definitions and Periodic Properties
    §20.2(iii) Translation of the Argument by Half-Periods
    5: Software Index
    6: Bibliography S
  • J. Segura and A. Gil (1998) Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.
  • 7: Bibliography B
  • L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1988a) Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.
  • 8: 14.34 Software
    §14.34(ii) Legendre Functions: Real Argument and Parameters
  • Gil and Segura (1997). Integer and half-integer parameters. Fortran.

  • §14.34(iii) Legendre Functions: Complex Argument and/or Parameters
  • Gil and Segura (1998). Integer parameters and purely imaginary arguments. Fortran.

  • 9: 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. …
  • Searches for resonances as poles of the S -matrix in the complex half-plane k < 0 . See for example Csótó and Hale (1997).

  • 10: 34.5 Basic Properties: 6 j Symbol
    If any lower argument in a 6 j symbol is 0 , 1 2 , or 1 , then the 6 j symbol has a simple algebraic form. … The 6 j symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example, …
    34.5.13 E ( j ) = ( ( j 2 - ( j 2 - j 3 ) 2 ) ( ( j 2 + j 3 + 1 ) 2 - j 2 ) ( j 2 - ( l 2 - l 3 ) 2 ) ( ( l 2 + l 3 + 1 ) 2 - j 2 ) ) 1 2 .
    34.5.19 l { j 1 j 2 l j 2 j 1 j } = 0 , 2 μ - j odd, μ = min ( j 1 , j 2 ) ,
    34.5.20 l ( - 1 ) l + j { j 1 j 2 l j 1 j 2 j } = ( - 1 ) 2 μ 2 j + 1 , μ = min ( j 1 , j 2 ) ,