Legendre relation

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51: Bibliography C

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

Encodings:

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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External Links:

Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

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52: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

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Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.

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External Links:

ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§14.29.

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

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53: 23.15 Definitions

§23.15 Definitions

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23.15.1 $$q=\exp \left(-\pi \frac{K^{\prime }\left(k\right)}{K\left(k\right)}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: modulus

Permalink:

http://dlmf.nist.gov/23.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

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54: 29.2 Differential Equations

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29.2.8 $$\eta ={(e_1-e_3)}^{-1/2}(z-\mathrm{i}{K}^{\prime }),$$

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Defines:

$\eta$ (locally)

Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable, $k$: real parameter and $e_j$: constants

Permalink:

http://dlmf.nist.gov/29.2.E8

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Annotations for §29.2(ii), §29.2 and Ch.29

… ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

55: 14.19 Toroidal (or Ring) Functions

… ►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 …Most required properties of toroidal functions come directly from the results for $P_{\nu }^{\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$. … ►

14.19.5 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}\frac{\cosh \left(mt\right)}{{(\cosh \xi +\cosh t\sinh \xi )}^{n+(1/2)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

A&S Ref:

8.11.4 (modified and corrected)

Permalink:

http://dlmf.nist.gov/14.19.E5

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See also:

Annotations for §14.19(iii), §14.19 and Ch.14

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14.19.7 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}{𝑸}_{m-\frac{1}{2}}^n\left(\coth \xi \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E7

Encodings:

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See also:

Annotations for §14.19(v), §14.19 and Ch.14

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14.19.8 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(m-n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(m+n+\frac{1}{2}\right)}{\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}{P}_{m-\frac{1}{2}}^n\left(\coth \xi \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(v), §14.19 and Ch.14

56: 23.6 Relations to Other Functions

§23.6 Relations to Other Functions

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§23.6(i) Theta Functions

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§23.6(ii) Jacobian Elliptic Functions

… ►For relations to symmetric elliptic integrals see §19.25(vi). … ►

57: 22.20 Methods of Computation

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§22.20(vi) Related Functions

… ►For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …

58: 19.2 Definitions

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§19.2(ii) Legendre’s Integrals

… ►Legendre’s complementary complete elliptic integrals are defined via … ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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59: Bibliography

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Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:

ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt

Referenced by:

(25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.

Encodings:

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

Encodings:

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:

ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

Encodings:

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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60: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
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… ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …