in terms of Whittaker functions

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21: 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 $\eta =\frac{1}{2}\beta a^2$. … ►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. …

22: 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 $\nu$ is unrelated to the $\nu$ in §31.6.) … ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ 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).

23: 28.31 Equations of Whittaker–Hill and Ince

§28.31 Equations of Whittaker–Hill and Ince

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§28.31(i) Whittaker–Hill Equation

►Hill’s equation with three terms …and constant values of $A,B,k$, and $c$, is called the Equation of Whittaker–Hill. … ►in (28.31.1). …

24: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation …The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. … ►Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. …

25: 3.10 Continued Fractions

… ►if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►The $A_n$ and $B_n$ of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). …

26: Bibliography L

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Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions. J. Electromagn. Waves Appl. 12 (6), pp. 709–711.

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External Links:

ISSN 0920-5071, Document, MathReview

Referenced by:

§30.14(v).

Encodings:

BibTeX

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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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External Links:

ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

Encodings:

BibTeX

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Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

Encodings:

BibTeX

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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

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External Links:

ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

BibTeX

…

27: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The negative-energy functions are widely used in the description of atomic and molecular spectra; see Bethe and Salpeter (1977), Seaton (1983), and Aymar et al. (1996). … ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. … ►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho /(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))$ (Mott and Massey (1956, pp. 63–65)). … ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. …

28: 22.2 Definitions

§22.2 Definitions

►The nome $q$ is given in terms of the modulus $k$ by … ►The Jacobian functions are related in the following way. … ►The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$. ►In terms of Neville’s theta functions (§20.1) …

29: 13.24 Series

§13.24 Series

►

§13.24(i) Expansions in Series of Whittaker Functions

►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). ►

§13.24(ii) Expansions in Series of Bessel Functions

… ►Additional expansions in terms of Bessel functions are given in Luke (1959). …

30: 23.2 Definitions and Periodic Properties

… ►

§23.2(i) Lattices

… ► … ►

§23.2(ii) Weierstrass Elliptic Functions

… ►Hence the order of the terms or factors is immaterial. … ► …