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11: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
12: 23.4 Graphics
§23.4(i) Real Variables
Line graphs of the Weierstrass functions ( x ) , ζ ( x ) , and σ ( x ) , illustrating the lemniscatic and equianharmonic cases. …
See accompanying text
Figure 23.4.6: σ ( x ; 0 , g 3 ) for 5 x 5 , g 3 = 0. … Magnify
§23.4(ii) Complex Variables
Surfaces for the Weierstrass functions ( z ) , ζ ( z ) , and σ ( z ) . …
13: William P. Reinhardt
  • Reinhardt was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …
    14: 22.1 Special Notation
    The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . … Other notations for sn ( z , k ) are sn ( z | m ) and sn ( z , m ) with m = k 2 ; see Abramowitz and Stegun (1964) and Walker (1996). …
    15: 20.9 Relations to Other Functions
    §20.9(ii) Elliptic Functions and Modular Functions
    See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
    16: 20.11 Generalizations and Analogs
    §20.11(ii) Ramanujan’s Theta Function and q -Series
    §20.11(iii) Ramanujan’s Change of Base
    As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). …This is Jacobi’s inversion problem of §20.9(ii). …
    17: Errata
  • Subsection 17.9(iii)

    The title of the paragraph which was previously “Gasper’s q -Analog of Clausen’s Formula” has been changed to “Gasper’s q -Analog of Clausen’s Formula (16.12.2)”.

  • Subsection 19.25(vi)

    This subsection has been significantly updated. In particular, the following formulae have been corrected. Equation (19.25.35) has been replaced by

    19.25.35 z + 2 ω = ± R F ( ( z ) e 1 , ( z ) e 2 , ( z ) e 3 ) ,

    in which the left-hand side z has been replaced by z + 2 ω for some 2 ω 𝕃 , and the right-hand side has been multiplied by ± 1 . Equation (19.25.37) has been replaced by

    19.25.37 ζ ( z + 2 ω ) + ( z + 2 ω ) ( z ) = ± 2 R G ( ( z ) e 1 , ( z ) e 2 , ( z ) e 3 ) ,

    in which the left-hand side ζ ( z ) + z ( z ) has been replaced by ζ ( z + 2 ω ) + ( z + 2 ω ) ( z ) and the right-hand side has been multiplied by ± 1 . Equation (19.25.39) has been replaced by

    19.25.39 ζ ( ω j ) + ω j e j = 2 R G ( 0 , e j e k , e j e ) ,

    in which the left-hand side η j was replaced by ζ ( ω j ) , for some 2 ω j 𝕃 and ( ω j ) = e j . Equation (19.25.40) has been replaced by

    19.25.40 z + 2 ω = ± σ ( z ) R F ( σ 1 2 ( z ) , σ 2 2 ( z ) , σ 3 2 ( z ) ) ,

    in which the left-hand side z has been replaced by z + 2 ω , and the right-hand side was multiplied by ± 1 . For more details see §19.25(vi).

  • 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.

  • Equation (23.2.4)
    23.2.4 ( z ) = 1 z 2 + w 𝕃 { 0 } ( 1 ( z w ) 2 1 w 2 )

    Originally the denominator ( z w ) 2 was given incorrectly as ( z w 2 ) .

    Reported 2012-02-16 by James D. Walker.

  • 18: 23.5 Special Lattices
    §23.5(ii) Rectangular Lattice
    In this case the lattice roots e 1 , e 2 , and e 3 are real and distinct. …
    §23.5(iii) Lemniscatic Lattice
    §23.5(iv) Rhombic Lattice
    §23.5(v) Equianharmonic Lattice
    19: Peter L. Walker
    Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. 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. …
  • 20: Software Index