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1: 18.39 Applications in the Physical Sciences
where x is a spatial coordinate, m the mass of the particle with potential energy V ( x ) , = h / ( 2 π ) is the reduced Planck’s constant, and ( a , b ) a finite or infinite interval. Here the term 2 2 m 2 x 2 represents the quantum kinetic energy of a single particle of mass m , and V ( x ) its potential energy. … and = k = m = 1 , has eigenfunctions … , = m e = e 2 = 4 π ϵ 0 = 1 , Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of a . u . to SI units is spelled out. … Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …
2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
A complex linear vector space V is called an inner product space if an inner product u , v is defined for all u , v V with the properties: (i) u , v is complex linear in u ; (ii) u , v = v , u ¯ ; (iii) v , v 0 ; (iv) if v , v = 0 then v = 0 . …Two elements u and v in V are orthogonal if u , v = 0 . … Functions f , g L 2 ( X , d α ) for which f g , f g = 0 are identified with each other. … , u λ , u λ = 0 , for λ λ . … The adjoint T of T does satisfy T f , g = f , T g where f , g = a b f ( x ) g ( x ) d x . …
3: Bibliography I
  • IEEE (2015) IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..
  • IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..
  • Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J 0 ( z ) i J 1 ( z ) and of Bessel functions J m ( z ) of any real order m . Linear Algebra Appl. 194, pp. 35–70.
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • 4: 20 Theta Functions
    Chapter 20 Theta Functions
    5: 27.2 Functions
    Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes p x : …
    Table 27.2.2: Functions related to division.
    n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
    5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
    6 2 4 12 19 18 2 20 32 16 6 63 45 24 6 78
    7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
    6: 18.40 Methods of Computation
    Let x ( a , b ) . …Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … Here x ( t , N ) is an interpolation of the abscissas x i , N , i = 1 , 2 , , N , that is, x ( i , N ) = x i , N , allowing differentiation by i . … This is a challenging case as the desired w RCP ( x ) on [ 1 , 1 ] has an essential singularity at x = 1 . … Further, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
    7: Bibliography L
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • H. Levine and J. Schwinger (1948) On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.
  • Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.
  • J. N. Lyness (1985) Integrating some infinite oscillating tails. J. Comput. Appl. Math. 12/13, pp. 109–117.
  • 8: 1.2 Elementary Algebra
    §1.2(iii) Partial Fractions
    1.2.41 𝐮 , 𝐯 = 𝐯 , 𝐮 ¯ ,
    1.2.43 𝐯 , 𝐯 = 0 ,
    1.2.44 𝐮 , 𝐯 = 0 .
    If det ( 𝐀 ) = 0 then, depending on 𝐜 , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of 𝐀 𝐛 = 𝟎 . …
    9: 25.12 Polylogarithms
    The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). …
    25.12.8 n = 1 cos ( n θ ) n 2 = π 2 6 π θ 2 + θ 2 4 .
    See accompanying text
    Figure 25.12.1: Dilogarithm function Li 2 ( x ) , 20 x < 1 . Magnify
    See accompanying text
    Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
    25.12.10 Li s ( z ) = n = 1 z n n s .
    10: 26.9 Integer Partitions: Restricted Number and Part Size
    Table 26.9.1: Partitions p k ( n ) .
    n k
    8 0 1 5 10 15 18 20 21 22 22 22
    It is also equal to the number of lattice paths from ( 0 , 0 ) to ( m , k ) that have exactly n vertices ( h , j ) , 1 h m , 1 j k , above and to the left of the lattice path. … where the inner sum is taken over all positive divisors of t that are less than or equal to k . …