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1: Peter L. Walker
Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
  • 2: 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. …
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    3: 22.3 Graphics
    §22.3(i) Real Variables: Line Graphs
    Line graphs of the functions sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , cd ( x , k ) , sd ( x , k ) , nd ( x , k ) , dc ( x , k ) , nc ( x , k ) , sc ( x , k ) , ns ( x , k ) , ds ( x , k ) , and cs ( x , k ) for representative values of real x and real k illustrating the near trigonometric ( k = 0 ), and near hyperbolic ( k = 1 ) limits. … sn ( x , k ) , cn ( x , k ) , and dn ( x , k ) as functions of real arguments x and k . …
    §22.3(iii) Complex z ; Real k
    §22.3(iv) Complex k
    4: Software Index
    Open Source With Book Commercial
    22 Jacobian Elliptic Functions
    ‘✓’ 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. …
  • 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.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    5: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.
  • E. H. Neville (1951) Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 6: Bibliography M
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • L. M. Milne-Thomson (1950) Jacobian Elliptic Function Tables. Dover Publications Inc., New York.
  • D. Mumford (1984) Tata Lectures on Theta. II. Birkhäuser Boston Inc., Boston, MA.
  • 7: 20.11 Generalizations and Analogs
    §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). … … 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).
    §20.11(iv) Theta Functions with Characteristics
    8: 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.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • A. Weil (1999) Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.
  • 9: 20.7 Identities
    §20.7(i) Sums of Squares
    §20.7(v) Watson’s Identities
    §20.7(vi) Landen Transformations
    §20.7(vii) Derivatives of Ratios of Theta Functions
    See Lawden (1989, pp. 19–20). …
    10: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • G. W. Spenceley and R. M. Spenceley (1947) Smithsonian Elliptic Functions Tables. Smithsonian Miscellaneous Collections, v. 109 (Publication 3863), The Smithsonian Institution, Washington, D.C..
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.