cylindrical polar coordinates

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21: 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

… ►

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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22: 31.17 Physical Applications

… ►Introduce elliptic coordinates $z_1$ and $z_2$ on $S_2$. Then ►

31.17.2 $$\frac{x_s^2}{z_k}+\frac{x_t^2}{z_k-1}+\frac{x_u^2}{z_k-a}=0,$$ $k=1,2$,

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Symbols:

$z$: complex variable, $a$: complex parameter and $x_s$, $x_t$, $x_u$: coordinates

Permalink:

http://dlmf.nist.gov/31.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

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23: 14.19 Toroidal (or Ring) Functions

… ►

§14.19(i) Introduction

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta ,\theta ,\varphi )$, which are related to Cartesian coordinates $(x,y,z)$ by …

24: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

25: Bibliography T

… ►

On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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External Links:

ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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26: 9.16 Physical Applications

… ►The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). …

27: 1.15 Summability Methods

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and …

28: 19.26 Addition Theorems

… ►

19.26.3 $$\sqrt{z}=\frac{\xi {\zeta }^{\prime }+{\eta }^{\prime }\zeta -\xi {\eta }^{\prime }}{\sqrt{\xi \eta {\zeta }^{\prime }}+\sqrt{{\xi }^{\prime }{\eta }^{\prime }\zeta }},$$

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Symbols:

$z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/19.26.E3

Encodings:

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See also:

Annotations for §19.26(i), §19.26 and Ch.19

►where … ►

19.26.14 $$(p-y)R_C(x,p)+(q-y)R_C(x,q)=(\eta -\xi )R_C(\xi ,\eta ),$$ $x\ge 0$, $y\ge 0$; $p,q\in ℝ\setminus \{0\}$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$: element of, $ℝ$: real line, $\setminus$: set subtraction, $x$: positive, $y$: positive, $\xi$: coordinate and $\eta$: coordinate

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E14

Encodings:

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See also:

Annotations for §19.26(i), §19.26 and Ch.19

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19.26.24 $$z={(\xi \zeta +\eta \zeta -\xi \eta )}^2/(4\xi \eta \zeta ),$$ $(\xi ,\eta ,\zeta )=(x+\lambda ,y+\lambda ,z+\lambda )$,

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Symbols:

$\lambda$: positive, $x$: positive, $y$: positive, $z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/19.26.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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29: 30.1 Special Notation

… ►

30: 3.5 Quadrature

… ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. …