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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
    See accompanying text
    Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
    See accompanying text
    Figure 22.3.27: Density plot of | sn ( 10 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
    See accompanying text
    Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
    See accompanying text
    Figure 22.3.29: Density plot of | sn ( 30 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
    4: 23 Weierstrass Elliptic and Modular
    Functions
    Chapter 23 Weierstrass Elliptic and Modular Functions
    5: Software Index
    Open Source With Book Commercial
    20 Theta 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. …
    6: 36.2 Catastrophes and Canonical Integrals
    36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
    36.2.29 Ψ ( H ) ( 0 , 0 , z ) = Ψ ( H ) ( 0 , 0 , z ) ¯ = 2 1 / 3 3 exp ( 1 27 i z 3 ) Ψ ( E ) ( 0 , 0 , z 2 2 / 3 ) , < z < .
    7: 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.
  • 8: Bibliography R
  • H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.
  • H. Rosengren (2004) Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.
  • R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.
  • 9: 36.4 Bifurcation Sets
    x = 9 20 z 2 .
    x = 3 20 z 2 ,
    Elliptic umbilic bifurcation set (codimension three): for fixed z , the section of the bifurcation set is a three-cusped astroid … Elliptic umbilic cusp lines (ribs): …
    §36.4(ii) Visualizations
    10: 19.36 Methods of Computation
    §19.36 Methods of Computation
    If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … Lee (1990) compares the use of theta functions for computation of K ( k ) , E ( k ) , and K ( k ) E ( k ) , 0 k 2 1 , with four other methods. …For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …