About the Project

relations to Jacobian elliptic functions

AdvancedHelp

(0.019 seconds)

11—20 of 28 matching pages

11: 22.16 Related Functions
12: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. … Similar expansions for cn 2 ( z , k ) and dn 2 ( z , k ) follow immediately from (22.6.1). … A related hyperbolic series is …
13: 20.11 Generalizations and Analogs
As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions22.2), and Weierstrass elliptic functions23.6(ii)) can be expanded in q -series via (20.9.1). However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). … For applications to rapidly convergent expansions for π see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …
14: 29.6 Fourier Series
§29.6 Fourier Series
With ϕ = 1 2 π am ( z , k ) , as in (29.2.5), we have … In addition, if H satisfies (29.6.2), then (29.6.3) applies. … Consequently, 𝐸𝑐 ν 2 m ( z , k 2 ) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … Here dn ( z , k ) is as in §22.2, and …
15: 22.20 Methods of Computation
§22.20 Methods of Computation
§22.20(iii) Landen Transformations
§22.20(iv) Lattice Calculations
§22.20(v) Inverse Functions
16: William P. Reinhardt
He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …
  • 17: 23.21 Physical Applications
    §23.21 Physical Applications
    In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 x 2 ) ( 1 k 2 x 2 ) . …
    §23.21(ii) Nonlinear Evolution Equations
    Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. …
    §23.21(iii) Ellipsoidal Coordinates
    18: 22.8 Addition Theorems
    §22.8 Addition Theorems
    §22.8(iii) Special Relations Between Arguments
    For these and related identities see Copson (1935, pp. 415–416). If sums/differences of the z j ’s are rational multiples of K ( k ) , then further relations follow. …
    19: Bibliography W
  • E. L. Wachspress (2000) Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.
  • P. L. Walker (2003) The analyticity of Jacobian functions with respect to the parameter k . Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
  • P. L. Walker (2009) The distribution of the zeros of Jacobian elliptic functions with respect to the parameter k . Comput. Methods Funct. Theory 9 (2), pp. 579–591.
  • A. Weil (1999) Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.
  • J. Wimp (1985) Some explicit Padé approximants for the function Φ / Φ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.
  • 20: Software Index
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Commercial Software.

    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.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.