associated%20Legendre%20equation

SearchAdvancedHelp

(0.001 seconds)

1—10 of 11 matching pages

1: Bibliography O

… ►

Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\mathrm{VI}}$ . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

ⓘ

External Links:

ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).

Encodings:

BibTeX

… ►

An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

… ►

Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.

ⓘ

Notes:

Includes single, double precision Fortran code.

External Links:

GAMS (module DXNRMP; package SLATEC), ISSN 0021-9991, Document, MathReview (G. P. Bhattacharjee), ZentralBlatt

Referenced by:

§14.32, 7th item.

Encodings:

BibTeX

… ►

Bessel Functions. Part III: Zeros and Associated Values. Royal Society Mathematical Tables, Volume 7, Cambridge University Press, Cambridge-New York.

ⓘ

External Links:

ISBN 0-521-06153-9, MathReview (L. Fox), ZentralBlatt

Referenced by:

§10.21(vi), §10.21(viii), §10.58, §10.74(vi), 1st item, 2nd item.

Encodings:

BibTeX

… ►

Legendre functions with both parameters large. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 175–185.

ⓘ

External Links:

FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§14.15(v), §14.15(v).

Encodings:

BibTeX

…

2: Bibliography V

… ►

Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

Notes:

Associated computer program available online

External Links:

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

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

… ►

A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

ⓘ

External Links:

MathReview (J. V. Wehausen), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§14.29.

Encodings:

BibTeX

… ►

Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

… ►

On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

…

3: Bibliography

… ►

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

ⓘ

External Links:

Document, MathReview (Mark Adler)

Referenced by:

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

Encodings:

BibTeX

… ►

SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

ⓘ

Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

Referenced by:

1st item.

Encodings:

BibTeX

►

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

ⓘ

External Links:

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

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.43(iii).

Encodings:

BibTeX

… ►

Associated Laguerre and Hermite polynomials. Proc. Roy. Soc. Edinburgh 96A, pp. 15–37.

ⓘ

External Links:

MathReview Entry

Referenced by:

(18.30.16), (18.30.17), (18.30.18), (18.30.19), (18.30.20), §18.30(iii), §18.30(iv), §18.30, §18.30(v), (18.35.10).

Encodings:

BibTeX

…

4: Bibliography C

… ►

The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.

ⓘ

External Links:

ISSN 0951-7715, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

… ►

The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

… ►

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

ⓘ

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).

Encodings:

BibTeX

… ►

On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

ⓘ

External Links:

ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

… ►

Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

…

5: Software Index

… ► ►►►►

	Open Source													With Book						Commercial
…
14 Legendre and Related Functions
…
20 Theta Functions
…

… ►

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►

Software associated with papers published in the journal Computer Physics Communications.

…

6: Bibliography B

… ►

On the Associated Legendre Polynomials. In Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 43–50.

ⓘ

External Links:

MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.30.

Encodings:

BibTeX

… ►

Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.

ⓘ

External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt

Referenced by:

§14.17(iii), §14.17(iii).

Encodings:

BibTeX

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

… ►

Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 16S

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

…

7: Bibliography L

… ►

Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

BibTeX

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

… ►

Essai sur la Théorie des Nombres. 2nd edition, Courcier, Paris.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§27.2(i).

Encodings:

BibTeX

►

Traité des fonctions elliptiques et des intégrales Eulériennes. Huzard-Courcier, Paris.

ⓘ

Notes:

vol.1, 1825; vol.2, 1826; three supplements, 1828–1832.

Referenced by:

§19.14(ii), Ch.19.

Encodings:

BibTeX

… ►

Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

…

8: Errata

… ►

The Legendre polynomial $P_n$ was mistakenly identified as the associated Legendre function $P_n$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

►

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

… ►

Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by $\{-\nu -1,\nu \}$. These were replaced by $\{\nu +1,-\nu \}$.

Reported by Hans Volkmer on 2019-01-30

… ►

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

… ►

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

…

9: Bibliography D

… ►

Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 25D

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

ⓘ

External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

BibTeX

►

Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.15(iii), §2.8(i).

Encodings:

BibTeX

…

10: Bibliography S

… ►

Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

ⓘ

Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

Encodings:

BibTeX

… ►

The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

ⓘ

External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.17(i).

Encodings:

BibTeX

… ►

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

ⓘ

External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

►

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

ⓘ

External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

►

On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

ⓘ

External Links:

ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX