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1: 13.28 Physical Applications

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§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 …

2: 28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ► ►

§28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …

3: 12.17 Physical Applications

§12.17 Physical Applications

… ►in Cartesian coordinates $x,y,z$ of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … …

4: 28.31 Equations of Whittaker–Hill and Ince

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§28.31(iii) Paraboloidal Wave Functions

►With (28.31.10) and (28.31.11), …are called paraboloidal wave functions. … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by … ►

Asymptotic Behavior

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5: Bibliography U

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Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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6: 36.5 Stokes Sets

… ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates … ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.10 $$160u^6+40u^4=Y^2.$$

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Symbols:

$Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►With coordinates … ►

36.5.17 $$Y_{\mathrm{S}}(X)=Y(u,|X|),$$

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Symbols:

$X$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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7: 20 Theta Functions

Chapter 20 Theta Functions

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8: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►when transformed to sphero-conal coordinates $r,\beta ,\gamma$: … ►

29.18.4 $$u(r,\beta ,\gamma )=u_1(r)u_2(\beta )u_3(\gamma ),$$

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Symbols:

$u$: solution, $r$: sphero-conal coordinate, $\beta$: sphero-conal coordinate, $\gamma$: sphero-conal coordinate and $u_j$: solutions

Permalink:

http://dlmf.nist.gov/29.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(i), §29.18 and Ch.29

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§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: …

9: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

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10: 26.2 Basic Definitions

… ►A k-dimensional lattice path is a directed path composed of segments that connect vertices in ${\{0,1,2,\mathrm{\dots }\}}^k$ so that each segment increases one coordinate by exactly one unit. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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3	3	20	627	37	21637
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