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1: 31.2 Differential Equations
Weierstrass’s Form
2: 23.21 Physical Applications
The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x - 4 x 3 . … Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
3: 29.2 Differential Equations
we have …
4: 23.9 Laurent and Other Power Series
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as 1 / ( z ) 0 . …
5: 23.20 Mathematical Applications
§23.20 Mathematical Applications
§23.20(i) Conformal Mappings
§23.20(iii) Factorization
§23.20(v) Modular Functions and Number Theory
6: Errata
  • Equation (17.9.3)


    17.9.3
    ϕ 1 2 ( a , b c ; q , z ) = ( a b z / c ; q ) ( b z / c ; q ) ϕ 2 3 ( a , c / b , 0 c , c q / ( b z ) ; q , q ) + ( a , b z , c / b ; q ) ( c , z , c / ( b z ) ; q ) ϕ 2 3 ( z , a b z / c , 0 b z , b z q / c ; q , q )

    Originally, the second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ϕ 1 2 is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).

    Reported by Roberto S. Costas-Santos on 2019-04-26

  • Equation (23.12.2)


    23.12.2
    ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) - 8 ( z - ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) )

    Originally, the factor of 2 was missing from the denominator of the argument of the cot function.

    Reported by Blagoje Oblak on 2019-05-27

  • Other Changes


    • The factor on the right-hand side of Equation (10.9.26) containing cos ( μ - ν ) θ has been been replaced with cos ( ( μ - ν ) θ ) to clarify the meaning.

    • In Paragraph Confluent Hypergeometric Functions in §10.16, several Whittaker confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

    • In Equation (15.6.9), it was clarified that λ .

    • Originally Equation (19.16.9) had the constraint a , a > 0 . This constraint was replaced with b 1 + + b n > a > 0 , b j . It therefore follows from Equation (19.16.10) that a > 0 . The last sentence of Subsection 19.16(ii) was elaborated to mention that generalizations may also be found in Carlson (1977b). These were suggested by Bastien Roucariès.

    • In Section 19.25(vi), the Weierstrass lattice roots e j , were labeled 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)). This was reported by Felix Ospald.

    • In 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.

    • In Equation (27.12.5), the term originally written as ln x was rewritten as ( ln x ) 1 / 2 to be consistent with other equations in the same subsection.

  • 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.

  • Table 22.5.4


    Originally the limiting form for sc ( z , k ) in the last line of this table was incorrect ( cosh z , instead of sinh z ).

    sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
    cn ( z , k ) sech z sd ( z , k ) sinh z nc ( z , k ) cosh z ds ( z , k ) csch z
    dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z

    Reported 2010-11-23.

  • 7: Software Index
    Open Source With Book Commercial
    23 Weierstrass Elliptic and Modular 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.

  • 8: 1.9 Calculus of a Complex Variable
    or in polar form ((1.9.3)) u and v satisfy … or its limiting form, and is invariant under bilinear transformations. …
    Weierstrass M -test
    9: 1.10 Functions of a Complex Variable
    A cut neighborhood is formed by deleting a ray emanating from the center. … It should be noted that different branches of ( w - w 0 ) 1 / μ used in forming ( w - w 0 ) n / μ in (1.10.16) give rise to different solutions of (1.10.12). …
    M -test
    Weierstrass Product