elliptical coordinates

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1: 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. … ► …

2: 31.17 Physical Applications

… ►Introduce elliptic coordinates

z_{1}

and

z_{2}

S_{2}

. Then …

3: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …In elliptical coordinates (28.32.2) becomes (28.32.3). …

4: 23.21 Physical Applications

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§23.21(iii) Ellipsoidal Coordinates

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5: 23.20 Mathematical Applications

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23.20.2

C:y^{2}z=x^{3}+axz^{2}+bz^{3},

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Symbols:: $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $z$ : complex, $C$ : elliptic curve, $a$ : constant and $b$ : constant
Permalink:: http://dlmf.nist.gov/23.20.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.20(ii), §23.20 and Ch.23

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6: 29.18 Mathematical Applications

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29.18.6

\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\beta,k\right))u_{2}=0,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\beta$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

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29.18.7

\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}=0,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

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7: 22.19 Physical Applications

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22.19.6

x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k\right).

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $k$ : modulus, $x(t)$ : coordinate, $t$ : time, $a$ : displacement and $\eta$ : parameter
Referenced by:: Erratum (V1.0.9) for Equation (22.19.6), Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)
Permalink:: http://dlmf.nist.gov/22.19.E6
Encodings:: pMML, png, TeX
Errata (effective with 1.0.9):: Originally the term $\sqrt{1+2\eta}$ was given incorrectly as $\sqrt{1+\eta}$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{2+\eta^{-1}}$ was given explicitly in the equation.
Reported 2014-04-28 by Svante Janson
See also:: Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.7

x(t)=a\operatorname{sn}\left(t\sqrt{1-\eta},k\right).

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $k$ : modulus, $x(t)$ : coordinate, $t$ : time, $a$ : displacement and $\eta$ : parameter
Referenced by:: Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)
Permalink:: http://dlmf.nist.gov/22.19.E7
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.9):: For added clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{\eta^{-1}-1}$ was given explicitly in the equation.
See also:: Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.8

x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right).

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $k$ : modulus, $x(t)$ : coordinate, $t$ : time, $a$ : displacement and $\eta$ : parameter
Referenced by:: Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)
Permalink:: http://dlmf.nist.gov/22.19.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.9

x(t)=a\operatorname{cn}\left(t\sqrt{2\eta-1},k\right),

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $k$ : modulus, $x(t)$ : coordinate, $t$ : time, $a$ : displacement and $\eta$ : parameter
Referenced by:: Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)
Permalink:: http://dlmf.nist.gov/22.19.E9
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.9):: For added clarity, the modulus $k$ has been defined in the line below. Previously $1/\sqrt{2-\eta^{-1}}$ was given explicitly in the equation.
See also:: Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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8: 22.18 Mathematical Applications

§22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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9: Bibliography R

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H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

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External Links:: Document
Referenced by:: 6th item.
Encodings:: BibTeX

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H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:: MathReview (H. H. Martens), ZentralBlatt
Referenced by:: §20.11(iv), §20.12(ii).
Encodings:: BibTeX

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W. Reinhardt (1982) Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annual Review of Physical Chemistry 33, pp. 223–255.

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Referenced by:: §1.18(viii).
Encodings:: BibTeX

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H. Rosengren (2004) Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §20.11(iii).
Encodings:: BibTeX

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R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:: ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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10: 36.5 Stokes Sets

… ►For

z\neq 0

, the Stokes set is expressed in terms of scaled coordinates … ►

§36.5(iii) Umbilics

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Elliptic Umbilic Stokes Set (Codimension three)

… ►With coordinates … ►

See accompanying text — Figure 36.5.5: Elliptic umbilic catastrophe with $z=\mathrm{constant}$ . … Magnify

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