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1: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
23.15.6 λ ( τ ) = θ 2 4 ( 0 , q ) θ 3 4 ( 0 , q ) ;
Dedekind’s Eta Function (or Dedekind Modular Function)
2: 27.14 Unrestricted Partitions
§27.14(iv) Relation to Modular Functions
Dedekind sums occur in the transformation theory of the Dedekind modular function η ( τ ) , defined by
27.14.12 η ( τ ) = e π i τ / 12 n = 1 ( 1 - e 2 π i n τ ) , τ > 0 .
For further properties of the function η ( τ ) see §§23.1523.19. …
27.14.16 Δ ( τ ) = ( 2 π ) 12 ( η ( τ ) ) 24 , τ > 0 ,
3: 23.16 Graphics
§23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions λ , J , and η . …
See accompanying text
Figure 23.16.1: Modular functions λ ( i y ) , J ( i y ) , η ( i y ) for 0 y 3 . … Magnify
See accompanying text
Figure 23.16.2: Elliptic modular function λ ( x + i y ) for - 0.25 x 0.25 , 0.005 y 0.1 . Magnify 3D Help
See accompanying text
Figure 23.16.3: Dedekind’s eta function η ( x + i y ) for - 0.0625 x 0.0625 , 0.0001 y 0.07 . Magnify 3D Help
4: 23.19 Interrelations
§23.19 Interrelations
23.19.1 λ ( τ ) = 16 ( η 2 ( 2 τ ) η ( 1 2 τ ) η 3 ( τ ) ) 8 ,
23.19.2 J ( τ ) = 4 27 ( 1 - λ ( τ ) + λ 2 ( τ ) ) 3 ( λ ( τ ) ( 1 - λ ( τ ) ) ) 2 ,
5: 23.1 Special Notation
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 ) . …
6: 23.17 Elementary Properties
§23.17 Elementary Properties
§23.17(i) Special Values
§23.17(ii) Power and Laurent Series
23.17.4 λ ( τ ) = 16 q ( 1 - 8 q + 44 q 2 + ) ,
§23.17(iii) Infinite Products
7: 23 Weierstrass Elliptic and Modular
Functions
Chapter 23 Weierstrass Elliptic and Modular Functions
8: 23.21 Physical Applications
§23.21 Physical Applications
§23.21(iv) Modular Functions
Physical applications of modular functions include: …
  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

  • 9: 23.18 Modular Transformations
    §23.18 Modular Transformations
    Elliptic Modular Function
    23.18.3 λ ( 𝒜 τ ) = λ ( τ ) ,
    23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,
    10: 23.20 Mathematical Applications
    For conformal mappings via modular functions see Apostol (1990, §2.7). …
    §23.20(iv) Modular and Quintic Equations
    §23.20(v) Modular Functions and Number Theory