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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 (23.6.15)
    23.6.15 σ ( u + ω j ) σ ( ω j ) = exp ( η j u + η 1 u 2 2 ω 1 ) θ j + 1 ( z , q ) θ j + 1 ( 0 , q ) , j = 1 , 2 , 3

    The factor exp ( η j u + η j u 2 2 ω 1 ) has been corrected to be exp ( η j u + η 1 u 2 2 ω 1 ) .

    Reported by Jan Felipe van Diejen on 2021-02-10

  • Section 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 Equation (19.25.35) has been replaced by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    in which the left-hand side z has been replaced by z + 2 ω for some 2 ω 𝕃 , and the right-hand side has been multiplied by ± 1 . Equation (19.25.37) has been replaced by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).

  • References

    Some references were added to §§7.25(ii), 7.25(iii), 7.25(vi), 8.28(ii), and to ¶Products (in §10.74(vii)) and §10.77(ix).

  • 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.