Whittaker%E2%80%93Hill%20equation

SearchAdvancedHelp

(0.003 seconds)

11—20 of 549 matching pages

11: 28.20 Definitions and Basic Properties

… ►

§28.20(i) Modified Mathieu’s Equation

►When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: ►

28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►

28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. …

12: 13.28 Physical Applications

… ►

§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). … ► ►

§13.28(ii) Coulomb Functions

…

13: 13.25 Products

§13.25 Products

►

13.25.1 $$M_{\kappa ,\mu }\left(z\right)M_{\kappa ,-\mu -1}\left(z\right)+\frac{(\frac{1}{2}+\mu +\kappa )(\frac{1}{2}+\mu -\kappa )}{4\mu (1+\mu ){(1+2\mu )}^2}{M}_{\kappa ,\mu +1}\left(z\right)M_{\kappa ,-\mu }\left(z\right)=1.$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.25

Permalink:

http://dlmf.nist.gov/13.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.25 and Ch.13

►For integral representations, integrals, and series containing products of $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).

14: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►

§13.20(i) Large $\mu$, Fixed $\kappa$

… ►Equations (13.20.17) and (13.20.18) are simpler than (6. … … ►For uniform approximations of $M_{\kappa ,\mathrm{i}\mu }\left(z\right)$ and $W_{\kappa ,\mathrm{i}\mu }\left(z\right)$, $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a).

15: 23 Weierstrass Elliptic and Modular
Functions

…

16: 13.15 Recurrence Relations and Derivatives

… ►

§13.15(i) Recurrence Relations

►

13.15.1 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -1,\mu }\left(z\right)+(z-2\kappa )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})M_{\kappa +1,\mu }\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.29

Permalink:

http://dlmf.nist.gov/13.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.2 $$2\mu (1+2\mu )\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-(z+2\mu )(1+2\mu )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})\sqrt{z}{M}_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.15(i)

Permalink:

http://dlmf.nist.gov/13.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.3 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+(1+2\mu )\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)-(\kappa +\mu +\frac{1}{2})M_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

… ►

§13.15(ii) Differentiation Formulas

…

17: Bibliography D

… ►

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

ⓘ

Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

External Links:

ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

… ►

Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

… ►

Uniform asymptotic approximations for the Whittaker functions $M_{\kappa ,i\mu }(z)$ and $W_{\kappa ,i\mu }(z)$ . Anal. Appl. (Singap.) 1 (2), pp. 199–212.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.20(v).

Encodings:

BibTeX

…

18: 20 Theta Functions

Chapter 20 Theta Functions

…

19: Bibliography

… ►

Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

… ►

Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

ⓘ

Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

… ►

Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-62321-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§1.10(vii), §1.15(iii), §1.15(vi), §1.15(vii), §10.22(vi), §13.6(v), §13.9(i), Ch.13, §14.3(iii), §14.30(iv), §15.10(i), §15.11(i), §15.11(ii), §15.14, §15.17(iv), §15.2(i), §15.4(ii), §15.5(i), §15.5(ii), §15.6, §15.7, §15.8(iv), §15.8(iii), Ch.15, §16.4(ii), §16.4(iii), §16.4(v), §16.4(v), Ch.16, §17.12, Ch.17, §18.10(ii), §18.10(iv), §18.11(i), §18.12, (18.17.41_5), §18.17(iv), §18.17(viii), §18.18(iv), §18.18(iv), §18.18(v), §18.18(vii), §18.2(i), §18.2(iv), §18.2(vi), §18.20(ii), §18.22(i), §18.25(ii), §18.26(i), Table 18.3.1, §18.38(ii), §18.5(iii), (18.7.25), Ch.18, §26.19, §4.10(i), §4.10(ii), §5.13, §5.14, §5.16, §5.16, §5.18(i), §5.18(ii), §5.18(iii), §5.20, §5.4(ii), §5.5(iv), Ch.5, Ch.6, Profile Richard A. Askey, Profile Ranjan Roy, Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

… ►

The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

ⓘ

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

… ►

Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

ⓘ

External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

…

20: 13.26 Addition and Multiplication Theorems

… ►

§13.26(i) Addition Theorems for $M_{\kappa ,\mu }\left(z\right)$

►The function $M_{\kappa ,\mu }\left(x+y\right)$ has the following expansions: … ►

§13.26(ii) Addition Theorems for $W_{\kappa ,\mu }\left(z\right)$

►The function $W_{\kappa ,\mu }\left(x+y\right)$ has the following expansions: … ►

§13.26(iii) Multiplication Theorems for $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$

…