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Dedekind eta function

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1: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions λ , J , and η . …
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Figure 23.16.1: Modular functions λ ( i y ) , J ( i y ) , η ( i y ) for 0 y 3 . … Magnify
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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
2: 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 ) . …
3: 23.19 Interrelations
23.19.1 λ ( τ ) = 16 ( η 2 ( 2 τ ) η ( 1 2 τ ) η 3 ( τ ) ) 8 ,
4: 23.15 Definitions
§23.15(ii) Functions λ ( τ ) , J ( τ ) , η ( τ )
Dedekind’s Eta Function (or Dedekind Modular Function)
5: 27.14 Unrestricted Partitions
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 .
27.14.14 η ( a τ + b c τ + d ) = ε ( - i ( c τ + d ) ) 1 2 η ( τ ) ,
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 ,
6: 20.3 Graphics
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Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. …Here q Dedekind = e - π y 0 = 0.19 approximately, where y = y 0 corresponds to the maximum value of Dedekind’s eta function η ( i y ) as depicted in Figure 23.16.1. Magnify
7: 23.17 Elementary Properties
23.17.6 η ( τ ) = n = - ( - 1 ) n q ( 6 n + 1 ) 2 / 12 .
23.17.8 η ( τ ) = q 1 / 12 n = 1 ( 1 - q 2 n ) ,
8: 23.18 Modular Transformations
Dedekind’s Eta Function
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,
9: Bibliography M
  • I. G. Macdonald (1972) Affine root systems and Dedekind’s η -function. Invent. Math. 15 (2), pp. 91–143.
  • 10: 23.22 Methods of Computation
    The modular functions λ ( τ ) , J ( τ ) , and η ( τ ) are also obtainable in a similar manner from their definitions in §23.15(ii). …