relation to Whittaker equation

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21: 28.32 Mathematical Applications

… ►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting

\zeta=\mathrm{i}\xi

z=\eta

in (28.32.3)). … ►

§28.32(ii) Paraboloidal Coordinates

… ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which

A, B

are separation constants. …

22: 5.17 Barnes’ $G$ -Function (Double Gamma Function)

§5.17 Barnes’ $G$ -Function (Double Gamma Function)

… ►In this equation (and in (5.17.5) below), the

\operatorname{Ln}

’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). ►When

z\to\infty

|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)

, ►

5.17.5

\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.

ⓘ

Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $B_{\NVar{n}}$ : Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $z$ : complex variable and $A$ : Glaisher’s constant
Referenced by:: §5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)
Permalink:: http://dlmf.nist.gov/5.17.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.1.11):: For consistency we have replaced $\operatorname{Ln}z$ by $\ln z$ .
Errata (effective with 1.0.7):: Originally the term $z\operatorname{Ln}\Gamma\left(z+1\right)$ was incorrectly stated as $z\Gamma\left(z+1\right)$ . (This error was reported subsequently by Nick Jones on December 11, 2013.)
Reported 2013-08-01 by Gergő Nemes
See also:: Annotations for §5.17 and Ch.5

… ►

5.17.7

C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer, $k$ : nonnegative integer and $C$ : log of Glaisher’s Constant
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

…

23: 22.8 Addition Theorems

… ►

§22.8(iii) Special Relations Between Arguments

►In the following equations the common modulus

k

is again suppressed. … ►A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530). … ►For these and related identities see Copson (1935, pp. 415–416). ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. …

24: Software Index

… ►

Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

►

Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►

Guide to Available Mathematical Software

A cross index of mathematical software in use at NIST.

25: Bibliography S

… ►

D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt
Referenced by:: §28.32(i).
Encodings:: BibTeX

… ►

S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

… ►

B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

ⓘ

External Links:: Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

… ►

F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when

p=q+1

. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

ⓘ

External Links:: Document, MathReview (C. S. Meijer), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

… ►

B. I. Suleĭmanov (1987) The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 23(1987), no. 5, pp. 569–576.
External Links:: ISSN 0374-0641, MathReview (T. Chanturiya), ZentralBlatt
Referenced by:: §32.11(ii).
Encodings:: BibTeX

…

26: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

… ►

20.5.14

\theta_{1}\left(z\middle|\tau\right)=z\theta_{1}'\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{\begin{subarray}{c}m=% -M\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.5.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §20.5(iii), §20.5 and Ch.20

►

20.5.15

\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+% \frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §20.5(iii), §20.5 and Ch.20

►

20.5.16

\theta_{3}\left(z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+% \frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §20.5(iii), §20.5 and Ch.20

►

20.5.17

\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+% \frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.5.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §20.5(iii), §20.5 and Ch.20

…

27: 9.6 Relations to Other Functions

§9.6 Relations to Other Functions

►

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

… ►

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

… ►

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

… ►To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).

28: Bibliography R

… ►

W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

ⓘ

External Links:: MathReview (E. A. Boyd), ZentralBlatt
Referenced by:: (9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).
Encodings:: BibTeX

►

W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

ⓘ

External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

►

W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

ⓘ

External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

R. R. Rosales (1978) The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

ⓘ

External Links:: FullText, MathReview (A. D. Brjuno), ZentralBlatt
Referenced by:: §32.11(ii), §32.17, §32.3(ii).
Encodings:: BibTeX

… ►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

ⓘ

External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

29: 23.2 Definitions and Periodic Properties

… ►

23.2.4

\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),

ⓘ

Defines:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function
Symbols:: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $z$ : complex
Keywords:: modular functions, Weierstrass $\wp$ -function
Referenced by:: §23.9, Erratum (V1.0.5) for Equation (23.2.4)
Permalink:: http://dlmf.nist.gov/23.2.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$ .
Reported 2012-02-16 by James D. Walker
See also:: Annotations for §23.2(ii), §23.2 and Ch.23

… ►When

z\notin\mathbb{L}

the functions are related by … ►When it is important to display the lattice with the functions they are denoted by

\wp\left(z|\mathbb{L}\right)

\zeta\left(z|\mathbb{L}\right)

, and

\sigma\left(z|\mathbb{L}\right)

, respectively. … ►

23.2.14

\eta_{3}\omega_{2}-\eta_{2}\omega_{3}=\eta_{2}\omega_{1}-\eta_{1}\omega_{2}=% \eta_{1}\omega_{3}-\eta_{3}\omega_{1}=\tfrac{1}{2}\pi i.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number
A&S Ref:: 18.3.37
Referenced by:: §23.12, §23.8(ii)
Permalink:: http://dlmf.nist.gov/23.2.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §23.2(iii), §23.2 and Ch.23

…

30: Bibliography W

… ►

E. J. Weniger (1996) 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.

ⓘ

External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

… ►

E. T. Whittaker (1902) On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §12.1, §12.5(i), §12.5(ii), §12.9(i).
Encodings:: BibTeX

►

E. T. Whittaker (1964) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.

ⓘ

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:: BibTeX

… ►

J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

ⓘ

Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

J. Wimp (1984) Computation with Recurrence Relations. Pitman, Boston, MA.

ⓘ

External Links:: ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt
Referenced by:: §13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).
Encodings:: BibTeX

…