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representation as Weierstrass elliptic functions

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1: 23.11 Integral Representations
§23.11 Integral Representations
2: 23.6 Relations to Other Functions
For representations of the Jacobi functions sn , cn , and dn as quotients of σ -functions see Lawden (1989, §§6.2, 6.3).
§23.6(iii) General Elliptic Functions
For representations of general elliptic functions23.2(iii)) in terms of σ ( z ) and ( z ) see Lawden (1989, §§8.9, 8.10), and for expansions in terms of ζ ( z ) see Lawden (1989, §8.11). …
3: Bibliography E
  • U. Eckhardt (1980) Algorithm 549: Weierstrasselliptic functions. ACM Trans. Math. Software 6 (1), pp. 112–120.
  • G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.
  • M. Eichler and D. Zagier (1982) On the zeros of the Weierstrass -function. Math. Ann. 258 (4), pp. 399–407.
  • T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.
  • J. A. Ewell (1990) A new series representation for ζ ( 3 ) . Amer. Math. Monthly 97 (3), pp. 219–220.
  • 4: Errata
  • Equation (11.11.1)

    Pochhammer symbol representations for the functions F k ( ν ) and G k ( ν ) were inserted.

  • Subsection 19.25(vi)

    The Weierstrass lattice roots e j , were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots e j , and lattice invariants g 2 , g 3 , now link to their respective definitions (see §§23.2(i), 23.3(i)).

    Reported by Felix Ospald.

  • Equation (19.25.37)

    The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

  • Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

    These equations were rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

  • Table 22.4.3

    Originally a minus sign was missing in the entries for cd u and dc u in the second column (headed z + K + i K ). The correct entries are - k - 1 ns z and - k sn z . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions sn , cn , dn , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

    u
    z + K z + K + i K z + i K z + 2 K z + 2 K + 2 i K z + 2 i K
    cd u - sn z - k - 1 ns z k - 1 dc z - cd z - cd z cd z
    dc u - ns z - k sn z k cd z - dc z - dc z dc z

    Reported 2014-02-28 by Svante Janson.