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11: Bibliography P
  • F. A. Paxton and J. E. Rollin (1959) Tables of the Incomplete Elliptic Integrals of the First and Third Kind. Technical report Curtiss-Wright Corp., Research Division, Quehanna, PA.
  • S. Pratt (2007) Comoving coordinate system for relativistic hydrodynamics. Phy. Rev. C 75, pp. (024907–1)–(024907–10).
  • W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.
  • 12: 19.26 Addition Theorems
    §19.26 Addition Theorems
    19.26.3 z = ξ ζ + η ζ ξ η ξ η ζ + ξ η ζ ,
    where …
    §19.26(iii) Duplication Formulas
    19.26.24 z = ( ξ ζ + η ζ ξ η ) 2 / ( 4 ξ η ζ ) , ( ξ , η , ζ ) = ( x + λ , y + λ , z + λ ) ,
    13: Bibliography H
  • M. H. Halley, D. Delande, and K. T. Taylor (1993) The combination of R -matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.
  • H. Hancock (1958) Elliptic Integrals. Dover Publications Inc., New York.
  • J. R. Herndon (1961a) Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.
  • J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.
  • 14: 36.6 Scaling Relations
    §36.6 Scaling Relations
    Indices for k -Scaling of Coordinates x m
    15: 36.7 Zeros
    §36.7(iii) Elliptic Umbilic Canonical Integral
    The zeros are lines in 𝐱 = ( x , y , z ) space where ph Ψ ( E ) ( 𝐱 ) is undetermined. …Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( 𝐱 ) | has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by …There are also three sets of zero lines in the plane z = 0 related by 2 π / 3 rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates ( x = r cos θ , y = r sin θ ) is given by …
    16: Errata
  • Equations (22.14.16), (22.14.17)
    22.14.16 0 K ( k ) ln ( sn ( t , k ) ) d t = π 4 K ( k ) 1 2 K ( k ) ln k ,
    22.14.17 0 K ( k ) ln ( cn ( t , k ) ) d t = π 4 K ( k ) + 1 2 K ( k ) ln ( k / k )

    Originally, a factor of π was missing from the terms containing the 1 4 K ( k ) .

    Reported by Fred Hucht on 2020-08-06

  • Table 22.4.3

    Originally a minus sign was missing in the entries for cd u and dc u in the second column (headed z + K + i K ). The correct entries are k 1 ns z and k sn z . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions sn , cn , dn , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

    u
    z + K z + K + i K z + i K z + 2 K z + 2 K + 2 i K z + 2 i K
    cd u sn z k 1 ns z k 1 dc z cd z cd z cd z
    dc u ns z k sn z k cd z dc z dc z dc z

    Reported 2014-02-28 by Svante Janson.

  • Equation (22.6.7)
    22.6.7 dn ( 2 z , k ) = dn 2 ( z , k ) k 2 sn 2 ( z , k ) cn 2 ( z , k ) 1 k 2 sn 4 ( z , k ) = dn 4 ( z , k ) + k 2 k 2 sn 4 ( z , k ) 1 k 2 sn 4 ( z , k )

    Originally the term k 2 sn 2 ( z , k ) cn 2 ( z , k ) was given incorrectly as k 2 sn 2 ( z , k ) dn 2 ( z , k ) .

    Reported 2014-02-28 by Svante Janson.

  • Table 22.5.4

    Originally the limiting form for sc ( z , k ) in the last line of this table was incorrect ( cosh z , instead of sinh z ).

    sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
    cn ( z , k ) sech z sd ( z , k ) sinh z nc ( z , k ) cosh z ds ( z , k ) csch z
    dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z

    Reported 2010-11-23.

  • Equation (36.10.14)
    36.10.14 3 ( 2 Ψ ( E ) x 2 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x x Ψ ( E ) = 0

    Originally this equation appeared with Ψ ( H ) x in the second term, rather than Ψ ( E ) x .

    Reported 2010-04-02.

  • 17: Bibliography M
  • J. N. Merner (1962) Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.
  • W. Miller (1974) Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.
  • L. M. Milne-Thomson (1950) Jacobian Elliptic Function Tables. Dover Publications Inc., New York.
  • P. Moon and D. E. Spencer (1971) Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions. 2nd edition, Springer-Verlag, Berlin.
  • T. Morita (1978) Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.