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21: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(i) Schrödinger Equation
§33.22(iv) Klein–Gordon and Dirac Equations
§33.22(vi) Solutions Inside the Turning Point
The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity ρ / ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) (Mott and Massey (1956, pp. 63–65)). …
  • Solution of relativistic Coulomb equations. See for example Cooper et al. (1979) and Barnett (1981b).

  • 22: 28.2 Definitions and Basic Properties
    For simple roots q of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations
    23: 30.8 Expansions in Series of Ferrers Functions
    Then the set of coefficients a n , k m ( γ 2 ) , k = - R , - R + 1 , - R + 2 , is the solution of the difference equation
    30.8.4 A k f k - 1 + ( B k - λ n m ( γ 2 ) ) f k + C k f k + 1 = 0 ,
    (note that A - R = 0 ) that satisfies the normalizing condition …
    30.8.8 λ n m ( γ 2 ) - B k A k a n , k m ( γ 2 ) a n , k - 1 m ( γ 2 ) = 1 + O ( 1 k 4 ) .
    The set of coefficients a n , k m ( γ 2 ) , k = - N - 1 , - N - 2 , , is the recessive solution of (30.8.4) as k - that is normalized by …
    24: 32.14 Combinatorics
    The distribution function F ( s ) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom (1994). … See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.
    25: 28.15 Expansions for Small q
    §28.15(i) Eigenvalues λ ν ( q )
    28.15.1 λ ν ( q ) = ν 2 + 1 2 ( ν 2 - 1 ) q 2 + 5 ν 2 + 7 32 ( ν 2 - 1 ) 3 ( ν 2 - 4 ) q 4 + 9 ν 4 + 58 ν 2 + 29 64 ( ν 2 - 1 ) 5 ( ν 2 - 4 ) ( ν 2 - 9 ) q 6 + .
    Higher coefficients can be found by equating powers of q in the following continued-fraction equation, with a = λ ν ( q ) :
    28.15.2 a - ν 2 - q 2 a - ( ν + 2 ) 2 - q 2 a - ( ν + 4 ) 2 - = q 2 a - ( ν - 2 ) 2 - q 2 a - ( ν - 4 ) 2 - .
    28.15.3 me ν ( z , q ) = e i ν z - q 4 ( 1 ν + 1 e i ( ν + 2 ) z - 1 ν - 1 e i ( ν - 2 ) z ) + q 2 32 ( 1 ( ν + 1 ) ( ν + 2 ) e i ( ν + 4 ) z + 1 ( ν - 1 ) ( ν - 2 ) e i ( ν - 4 ) z - 2 ( ν 2 + 1 ) ( ν 2 - 1 ) 2 e i ν z ) + ;
    26: Bibliography M
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
  • H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
  • H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.
  • J. P. Mills (1926) Table of the ratio: Area to bounding ordinate, for any portion of normal curve. Biometrika 18, pp. 395–400.
  • 27: 31.11 Expansions in Series of Hypergeometric Functions
    Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … Let w ( z ) be any Fuchs–Frobenius solution of Heun’s equation. …The coefficients c j satisfy the equationsEvery Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … In this case the accessory parameter q is a root of the continued-fraction equation
    28: Bibliography L
  • G. Labahn and M. Mutrie (1997) Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
  • D. W. Lozier and J. M. Smith (1981) Algorithm 567: Extended-range arithmetic and normalized Legendre polynomials [A1], [C1]. ACM Trans. Math. Software 7 (1), pp. 141–146.
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
  • 29: 22.18 Mathematical Applications
    §22.18(i) Lengths and Parametrization of Plane Curves
    §22.18(iii) Uniformization and Other Parametrizations
    The special case y 2 = ( 1 - x 2 ) ( 1 - k 2 x 2 ) is in Jacobian normal form. For any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) on this curve, their sum ( x 3 , y 3 ) , always a third point on the curve, is defined by the Jacobi–Abel addition law …
    30: 3.2 Linear Algebra
    With y = [ y 1 , y 2 , , y n ] T the process of solution can then be regarded as first solving the equation L y = b for y (forward elimination), followed by the solution of U x = y for x (back substitution). … A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with p = 2 . … where x and y are the normalized right and left eigenvectors of A corresponding to the eigenvalue λ . … Define the Lanczos vectors v j and coefficients α j and β j by v 0 = 0 , a normalized vector v 1 (perhaps chosen randomly), α 1 = v 1 T A v 1 , β 1 = 0 , and for j = 1 , 2 , , n - 1 by the recursive scheme …