Whittaker%20functions

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11: 28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►The approximations are expressed in terms of Whittaker functions $W_{\kappa ,\mu }\left(z\right)$ and $M_{\kappa ,\mu }\left(z\right)$ with $\mu =\frac{1}{4}$; 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). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

12: 28.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the Mathieu functions …and the modified Mathieu functions … ►Alternative notations for the functions are as follows. … ►

Abramowitz and Stegun (1964, Chapter 20)

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13: Bibliography O

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

Encodings:

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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.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

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Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

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External Links:

ISSN 0891-6837, MathReview Entry, ZentralBlatt

Referenced by:

§10.29(i), §10.6(i).

Encodings:

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14: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

… ►where $t(\zeta )$ is the function inverse to $\zeta (t)$, defined by (12.10.39) (see also (12.10.41)), and … ►For further information, including associated functions, see Olver (1959).

15: Bibliography W

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

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A Course of Modern Analysis. 4th edition, Cambridge University Press.

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Notes:

Reprinted in 1996. Table errata: Math. Comp. v. 36 (1981), no. 153, p. 319.

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.3(iii), §10.1, §12.11(i), §12.9(i), Ch.13, §15.6, Ch.15, Ch.19, §20.1, §20.1, §20.13, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.4(ii), §20.5(ii), §20.7(ii), §20.7(iv), §20.7(viii), §20.9(i), §20.9(ii), §20.9(ii), Ch.20, §22.11, §22.13(i), §22.13(ii), §22.14(i), §22.14(iii), §22.14(iv), §22.15(i), §22.16(ii), §22.16(iii), §22.18(i), §22.18(iv), §22.2, §22.4(i), §22.4(ii), §22.4(iii), §22.5(i), §22.6(i), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), §22.8(iii), §22.8(iii), Ch.22, §23.1, §23.10(i), §23.2(ii), §23.2(iii), §23.22(ii), §23.3(i), §23.3(ii), §23.6(i), Ch.23, §24.1, Table 28.1.1, §29.12(iii), Ch.29, §31.8, §31.8, Ch.4, §5.11(i), §5.11(ii), §5.17, §5.8, (5.9.10_2), §5.9(i), §5.9(i), §5.9(ii), Ch.5, Notations K.

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On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:

Document, ZentralBlatt

Referenced by:

§12.1, §12.5(i), §12.5(ii), §12.9(i).

Encodings:

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-521-35883-3, MathReview, ZentralBlatt

Referenced by:

§22.19(i), §22.19(iv), §22.19(v), §23.21(i).

Encodings:

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16: 18.39 Applications in the Physical Sciences

… ►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►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. … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►which corresponds to the exact results, in terms of Whittaker functions, of §§33.2 and 33.14, in the sense that projections onto the functions ${\varphi }_{n,l}(sr)/r$, the functions bi-orthogonal to ${\varphi }_{n,l}(sr)$, are identical. …