About the Project

normal equations

AdvancedHelp

(0.002 seconds)

21—30 of 74 matching pages

21: 28.31 Equations of Whittaker–Hill and Ince
§28.31 Equations of Whittaker–Hill and Ince
§28.31(i) Whittaker–Hill Equation
§28.31(ii) Equation of Ince; Ince Polynomials
The normalization is given by … They satisfy the differential equation
22: 36.10 Differential Equations
In terms of the normal form (36.2.1) the Ψ K ( 𝐱 ) satisfy the operator equationIn terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( 𝐱 ) satisfy the following operator equations
23: 1.13 Differential Equations
Equation (1.13.26) with x [ a , b ] may be transformed to the Liouville normal form
24: 9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
Equations of the form … In Olver (1977a, 1978) a different normalization is used. … Another normalization of (9.13.17) is used in Smirnov (1960), given by …
25: 3.7 Ordinary Differential Equations
The eigenvalues λ k are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …
26: 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).

  • 27: 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
    28: Errata
  • Equation (18.35.5)
    18.35.5 1 1 P n ( λ ) ( x ; a , b ) P m ( λ ) ( x ; a , b ) w ( λ ) ( x ; a , b ) d x = Γ ( 2 λ + n ) n ! ( λ + a + n ) δ n , m , a b a , λ > 0

    This equation was updated to give the full normalization. Previously the constraints on a , b and λ were given in (18.35.6) and included λ > 1 2 . The case 1 2 < λ 0 is now discussed in (18.35.6_2)–(18.35.6_4).

  • Chapter 18 Additions

    The following additions were made in Chapter 18:

    • Section 18.2

      In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)18.2(xii), with Equations (18.2.21)–(18.2.46),

    • Section 18.3

      A new introduction, minor changes in the presentation, and three new paragraphs.

    • Section 18.5

      Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

    • Section 18.8

      Line numbers and two extra rows were added to Table 18.8.1.

    • Section 18.9

      Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

    • Section 18.12

      Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

    • Section 18.14

      Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

    • Section 18.15

      Equation (18.15.4_5).

    • Section 18.16

      The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

    • Section 18.17

      Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

    • Section 18.18

      Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

    • Section 18.19

      A new introduction.

    • Section 18.21

      Equation (18.21.13).

    • Section 18.25

      Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

    • Section 18.26

      In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

    • Section 18.27

      Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

    • Section 18.28

      A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)18.28(xi), with Equations (18.28.23)–(18.28.34).

    • Section 18.30

      Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)18.30(viii), with Equations (18.30.8)–(18.30.31).

    • Section 18.32

      This short section has been expanded, with Equation (18.32.2).

    • Section 18.33

      Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

    • Section 18.34

      This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

    • Section 18.35

      This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

    • Section 18.36

      This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

    • Section 18.38

      The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

    • Section 18.39

      This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

      The original text of 18.39(i) Quantum Mechanics was:

      “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) :

      errata.1 ( 2 2 m 2 x 2 + V ( x ) ) ψ ( x , t ) = i t ψ ( x , t ) ,

      where is the reduced Planck’s constant. On substituting ψ ( x , t ) = η ( x ) ζ ( t ) , we obtain two ordinary differential equations, each of which involve the same constant E . The equation for η ( x ) is

      errata.2 d 2 η d x 2 + 2 m 2 ( E V ( x ) ) η = 0 .

      For a harmonic oscillator, the potential energy is given by

      errata.3 V ( x ) = 1 2 m ω 2 x 2 ,

      where ω is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval < x < , the constant E (the total energy of the particle) must satisfy

      errata.4 E = E n = ( n + 1 2 ) ω , n = 0 , 1 , 2 , .

      The corresponding eigenfunctions are

      errata.5 η n ( x ) = π 1 4 2 1 2 n ( n ! b ) 1 2 H n ( x / b ) e x 2 / 2 b 2 ,

      where b = ( / m ω ) 1 / 2 , and H n is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

      A second example is provided by the three-dimensional time-independent Schrödinger equation

      errata.6 2 ψ + 2 m 2 ( E V ( 𝐱 ) ) ψ = 0 ,

      when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

      For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

    • Section 18.40

      The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

  • Equation (10.22.72)
    10.22.72 0 J μ ( a t ) J ν ( b t ) J ν ( c t ) t 1 μ d t = ( b c ) μ 1 sin ( ( μ ν ) π ) ( sinh χ ) μ 1 2 ( 1 2 π 3 ) 1 2 a μ e ( μ 1 2 ) i π Q ν 1 2 1 2 μ ( cosh χ ) , μ > 1 2 , ν > 1 , a > b + c , cosh χ = ( a 2 b 2 c 2 ) / ( 2 b c )

    Originally, the factor on the right-hand side was written as ( b c ) μ 1 cos ( ν π ) ( sinh χ ) μ 1 2 ( 1 2 π 3 ) 1 2 a μ , which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre function of the second kind Q ν μ . Watson’s Q ν μ equals sin ( ( ν + μ ) π ) sin ( ν π ) e μ π i Q ν μ in the DLMF.

    Reported by Arun Ravishankar on 2018-10-22

  • 29: 18.35 Pollaczek Polynomials
    18.35.5 1 1 P n ( λ ) ( x ; a , b ) P m ( λ ) ( x ; a , b ) w ( λ ) ( x ; a , b ) d x = Γ ( 2 λ + n ) n ! ( λ + a + n ) δ n , m , a b a , λ > 0 ,
    30: 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 …