conical functions

SearchAdvancedHelp

(0.001 seconds)

1—10 of 15 matching pages

1: 14.20 Conical (or Mehler) Functions

… ►

§14.20(i) Definitions and Wronskians

… ► … ►

§14.20(ii) Graphics

… ►

►►

Figure 14.20.8: ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-2}\left(x\right)$, $\tau =0,\frac{1}{2},1,2,4$. Magnify

… ►

§14.20(x) Zeros and Integrals

…

2: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

3: 14.1 Special Notation

§14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

4: 14.33 Tables

… ►

Žurina and Karmazina (1964, 1965) tabulate the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

►

Žurina and Karmazina (1963) tabulates the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

…

5: 14.32 Methods of Computation

… ►

For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

6: 14.23 Values on the Cut

… ►The conical function defined by (14.20.2) can be represented similarly by ►

14.23.7 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{1}{2}{\mathrm{e}}^{3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x-\mathrm{i}0\right)+\frac{1}{2}{\mathrm{e}}^{-3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}-\mathrm{i}\tau }^{-\mu }\left(x+\mathrm{i}0\right).$$

ⓘ

Symbols:

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

7: Bibliography D

… ►

Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.

ⓘ

External Links:

ISSN 0308-2105, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.20(i), §14.20(ix), §14.20(ix), §14.20(viii), §14.23.

Encodings:

BibTeX

… ►

Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.

ⓘ

External Links:

ISSN 0308-2105, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§14.20(ix), §14.20(ix).

Encodings:

BibTeX

…

8: Bibliography G

… ►

Computing the conical function $P_{-1/2+i\tau }^{\mu }(x)$ . SIAM J. Sci. Comput. 31 (3), pp. 1716–1741.

ⓘ

External Links:

ISSN 1064-8275, Document, Link, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

… ►

An improved algorithm and a Fortran 90 module for computing the conical function $P_{-1/2+i\tau }^m(x)$ . Comput. Phys. Commun. 183 (3), pp. 794–799.

ⓘ

External Links:

ISSN 0010-4655, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

…

9: 3.8 Nonlinear Equations

… ►For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). …

10: Bibliography K

… ►

A Program for Computing the Conical Functions of the First Kind $P_{-1/2+i\tau }^m(x)$ for $m=0$ and $m=1$ . Comput. Phys. Comm. 23 (1), pp. 51–61.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 15D.

External Links:

Document, Code

Referenced by:

§14.31(ii), 1st item.

Encodings:

BibTeX

…