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in terms of Whittaker functions

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11: Bibliography O
  • F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
  • 12: 13.21 Uniform Asymptotic Approximations for Large κ
    For a uniform asymptotic expansion in terms of Airy functions for W κ , μ ( 4 κ x ) when κ is large and positive, μ is real with | μ | bounded, and x [ δ , ) see Olver (1997b, Chapter 11, Ex. 7.3). …
    13: 13.20 Uniform Asymptotic Approximations for Large μ
    §13.20(i) Large μ , Fixed κ
    It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. … These approximations are in terms of Airy functions. …
    14: 33.16 Connection Formulas
    §33.16(iii) f and h in Terms of W κ , μ ( z ) when ϵ < 0
    §33.16(v) s and c in Terms of W κ , μ ( z ) when ϵ < 0
    15: 22.14 Integrals
    With x , … For alternative, and symmetric, formulations of the results in this subsection see Carlson (2006a). … The indefinite integral of a 4th power can be expressed as a complete elliptic integral, a polynomial in Jacobian functions, and the integration variable. … For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b). …
    16: 5.11 Asymptotic Expansions
    The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k = n - 1 ( k 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. … For the remainder term in (5.11.3) write … For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … For the error term in (5.11.19) in the case z = x ( > 0 ) and c = 1 , see Olver (1995). …
    17: Errata
  • Subsections 8.18(ii)8.11(v)

    A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables a and x = λ a . It has been extended to allow for complex variables a and z = λ a (and we have replaced x with z in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from 0 < λ < 1 to the sector 0 < λ < 1 , | ph a | π 2 - δ , and the interval of validity of (8.11.7) was increased from λ > 1 to the sector λ > 1 , | ph a | 3 π 2 - δ . A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from | ph z | π - δ to | ph z | 2 π - δ . Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

  • 18: 20.4 Values at z = 0
    §20.4 Values at z = 0
    §20.4(i) Functions and First Derivatives
    20.4.2 θ 1 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) 3 = 2 q 1 / 4 ( q 2 ; q 2 ) 3 ,
    20.4.3 θ 2 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) ( 1 + q 2 n ) 2 ,
    Jacobi’s Identity
    19: 22.19 Physical Applications
    The subsequent position as a function of time, x ( t ) , for the three cases is given with results expressed in terms of a and the dimensionless parameter η = 1 2 β a 2 . … Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. … The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …Elementary discussions of this topic appear in Lawden (1989, §5.7), Greenhill (1959, pp. 101–103), and Whittaker (1964, Chapter VI). … Whittaker (1964, Chapter IV) enumerates the complete class of one-body classical mechanical problems that are solvable this way. …
    20: 31.8 Solutions via Quadratures
    the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … (This ν is unrelated to the ν in §31.6.) … By automorphisms from §31.2(v), similar solutions also exist for m 0 , m 1 , m 2 , m 3 , and Ψ g , N ( λ , z ) may become a rational function in z . …For m = ( m 0 , 0 , 0 , 0 ) , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. … The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).