relations to Jacobian elliptic functions

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21: Bibliography C

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Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

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Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.

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ISSN 0377-0427, Document, MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

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Some reformulated properties of Jacobian elliptic functions. J. Math. Anal. Appl. 323 (1), pp. 522–529.

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

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.13(ii), §22.13(iii), §22.14(i), §22.14(ii).

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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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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

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22: 29.14 Orthogonality

… ►First, the orthogonality relations (29.3.19) apply; see §29.12(i). Secondly, the system of functions …is orthogonal and complete with respect to the inner product …where … ►Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …

23: 20.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the theta functions ${\theta }_j\left(z\right|\tau )={\theta }_j(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. …Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. … ►This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). …

24: Bibliography B

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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1st item.

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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2nd item, 1st item.

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

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ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

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25: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

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26: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: …These functions are polynomials in $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$. … ►The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$. … ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

27: Bibliography N

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

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MathReview

Referenced by:

§19.39(iii).

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Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§20.1, Notations T, Notations T, Notations T, Notations T.

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28: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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FullText, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

3rd item.

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

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