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1: 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 23.16.3: Dedekind’s eta function η ( x + i y ) for - 0.0625 x 0.0625 , 0.0001 y 0.07 . Magnify 3D Help
2: 23.15 Definitions
§23.15(ii) Functions λ ( τ ) , J ( τ ) , η ( τ )
Dedekind’s Eta Function (or Dedekind Modular Function)
3: 23.19 Interrelations
23.19.1 λ ( τ ) = 16 ( η 2 ( 2 τ ) η ( 1 2 τ ) η 3 ( τ ) ) 8 ,
4: 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 ) . …
5: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.3 P ( a , z ) = 1 2 erfc ( - η a / 2 ) - S ( a , η ) ,
8.12.4 Q ( a , z ) = 1 2 erfc ( η a / 2 ) + S ( a , η ) ,
Then as a in the sector | ph a | π - δ ( < π ) ,
8.12.7 S ( a , η ) e - 1 2 a η 2 2 π a k = 0 c k ( η ) a - k ,
8.12.8 T ( a , η ) e 1 2 a η 2 2 π a k = 0 c k ( η ) ( - a ) - k ,
6: 33.11 Asymptotic Expansions for Large ρ
§33.11 Asymptotic Expansions for Large ρ
33.11.1 H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k ,
33.11.4 H ± ( η , ρ ) = e ± i θ ( f ( η , ρ ) ± i g ( η , ρ ) ) ,
33.11.7 g ( η , ρ ) f ^ ( η , ρ ) - f ( η , ρ ) g ^ ( η , ρ ) = 1 .
7: 33.2 Definitions and Basic Properties
§33.2(ii) Regular Solution F ( η , ρ )
F ( η , ρ ) is a real and analytic function of ρ on the open interval 0 < ρ < , and also an analytic function of η when - < η < . …
§33.2(iii) Irregular Solutions G ( η , ρ ) , H ± ( η , ρ )
Also, e i σ ( η ) H ± ( η , ρ ) are analytic functions of η when - < η < .
§33.2(iv) Wronskians and Cross-Product
8: 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 ) ,
9: 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 .
η ( τ ) satisfies the following functional equation: if a , b , c , d are integers with a d - b c = 1 and c > 0 , then … 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 ,
10: 23.18 Modular Transformations
Dedekind’s Eta Function
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,