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1: 1.10 Functions of a Complex Variable
M -test
Weierstrass Product
2: 23.8 Trigonometric Series and Products
§23.8(iii) Infinite Products
23.8.6 σ ( z ) = 2 ω 1 π exp ( η 1 z 2 2 ω 1 ) sin ( π z 2 ω 1 ) n = 1 1 - 2 q 2 n cos ( π z / ω 1 ) + q 4 n ( 1 - q 2 n ) 2 ,
23.8.7 σ ( z ) = 2 ω 1 π exp ( η 1 z 2 2 ω 1 ) sin ( π z 2 ω 1 ) n = 1 sin ( π ( 2 n ω 3 + z ) / ( 2 ω 1 ) ) sin ( π ( 2 n ω 3 - z ) / ( 2 ω 1 ) ) sin 2 ( π n ω 3 / ω 1 ) .
3: 23.1 Special Notation
𝕃 lattice in .
= e i π τ nome.
Δ discriminant g 2 3 - 27 g 3 2 .
G × H Cartesian product of groups G and H , that is, the set of all pairs of elements ( g , h ) with group operation ( g 1 , h 1 ) + ( g 2 , h 2 ) = ( g 1 + g 2 , h 1 + h 2 ) .
The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
4: 23.2 Definitions and Periodic Properties
23.2.6 σ ( z ) = z w 𝕃 { 0 } ( ( 1 - z w ) exp ( z w + z 2 2 w 2 ) ) .
5: 23.10 Addition Theorems and Other Identities
23.10.13 σ ( n z ) = A n e - n ( n - 1 ) ( η 1 + η 3 ) z j = 0 n - 1 = 0 n - 1 σ ( z + 2 j n ω 1 + 2 n ω 3 ) ,
23.10.14 A n = n j = 0 n - 1 = 0 j n - 1 1 σ ( ( 2 j ω 1 + 2 ω 3 ) / n ) .
6: 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
7: Errata
  • 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


  • 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.