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11: 28.34 Methods of Computation
§28.34(i) Characteristic Exponents
  • (c)

    Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

  • (d)

    Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • (d)

    Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

  • 12: 13.29 Methods of Computation
    Similarly for the Whittaker functions. … The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. … The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … normalizing relationnormalizing relation
    13: 20.9 Relations to Other Functions
    §20.9 Relations to Other Functions
    §20.9(i) Elliptic Integrals
    §20.9(ii) Elliptic Functions and Modular Functions
    The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
    §20.9(iii) Riemann Zeta Function
    14: 22.16 Related Functions
    §22.16 Related Functions
    Relation to Elliptic Integrals
    Relation to Theta Functions
    Relation to the Elliptic Integral E ( ϕ , k )
    Definition
    15: 28.8 Asymptotic Expansions for Large q
    For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … The approximations are expressed in terms of Whittaker functions W κ , μ ( z ) and M κ , μ ( z ) with μ = 1 4 ; compare §2.8(vi). …With additional restrictions on z , uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). … For related results see Langer (1934) and Sharples (1967, 1971). …
    16: Errata
  • Equation (18.34.1)
    18.34.1 y n ( x ; a ) = F 0 2 ( n , n + a 1 ; x 2 ) = ( n + a 1 ) n ( x 2 ) n F 1 1 ( n 2 n a + 2 ; 2 x ) = n ! ( 1 2 x ) n L n ( 1 a 2 n ) ( 2 x 1 ) = ( 1 2 x ) 1 1 2 a e 1 / x W 1 1 2 a , 1 2 ( a 1 ) + n ( 2 x 1 )

    This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.

  • Additions

    Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to F ( a , a ; a + 1 ; 1 2 ) , (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

  • Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

    The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

  • Equation (9.7.2)

    Following a suggestion from James McTavish on 2017-04-06, the recurrence relation u k = ( 6 k 5 ) ( 6 k 3 ) ( 6 k 1 ) ( 2 k 1 ) 216 k u k 1 was added to Equation (9.7.2).

  • Chapter 25 Zeta and Related Functions

    A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

  • 17: 33.2 Definitions and Basic Properties
    §33.2(i) Coulomb Wave Equation
    The function F ( η , ρ ) is recessive (§2.7(iii)) at ρ = 0 , and is defined by …where M κ , μ ( z ) and M ( a , b , z ) are defined in §§13.14(i) and 13.2(i), and … The functions H ± ( η , ρ ) are defined by …where W κ , μ ( z ) , U ( a , b , z ) are defined in §§13.14(i) and 13.2(i), …
    18: 33.14 Definitions and Basic Properties
    §33.14(i) Coulomb Wave Equation
    §33.14(ii) Regular Solution f ( ϵ , ; r )
    where M κ , μ ( z ) and M ( a , b , z ) are defined in §§13.14(i) and 13.2(i), and …
    §33.14(iii) Irregular Solution h ( ϵ , ; r )
    For nonzero values of ϵ and r the function h ( ϵ , ; r ) is defined by …
    19: 10.39 Relations to Other Functions
    §10.39 Relations to Other Functions
    Elementary Functions
    Parabolic Cylinder Functions
    Confluent Hypergeometric Functions
    Generalized Hypergeometric Functions and Hypergeometric Function
    20: 33.23 Methods of Computation
    The methods used for computing the Coulomb functions described below are similar to those in §13.29. …
    §33.23(iii) Integration of Defining Differential Equations
    §33.23(iv) Recurrence Relations
    In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer , provided that the recurrence is carried out in a stable direction (§3.6). … Noble (2004) obtains double-precision accuracy for W η , μ ( 2 ρ ) for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …