Klein complete invariant ♦ 6 matching pages ♦ SearchAdvancedHelp (0.001 seconds) 6 matching pages 1: 23.19 Interrelations … ► 23.19.2 J ( τ ) = 4 27 ( 1 − λ ( τ ) + λ 2 ( τ ) ) 3 ( λ ( τ ) ( 1 − λ ( τ ) ) ) 2 , ⓘ Symbols: J ( τ ) : Klein’s complete invariant, λ ( τ ) : elliptic modular function and τ : complex variable Permalink: http://dlmf.nist.gov/23.19.E2 Encodings: pMML, png, TeX See also: Annotations for §23.19 and Ch.23 ► 23.19.3 J ( τ ) = g 2 3 g 2 3 − 27 g 3 2 , ⓘ Symbols: J ( τ ) : Klein’s complete invariant, g j : Weierstrass lattice invariants g 2 , g 3 , 𝕃 : lattice and τ : complex variable Permalink: http://dlmf.nist.gov/23.19.E3 Encodings: pMML, png, TeX See also: Annotations for §23.19 and Ch.23 … 2: 23.15 Definitions … ► Klein’s Complete Invariant ► 23.15.7 J ( τ ) = ( θ 2 8 ( 0 , q ) + θ 3 8 ( 0 , q ) + θ 4 8 ( 0 , q ) ) 3 54 ( θ 1 ′ ( 0 , q ) ) 8 , ⓘ Defines: J ( τ ) : Klein’s complete invariant Symbols: θ j ( z , q ) : theta function, q : nome and τ : complex variable Permalink: http://dlmf.nist.gov/23.15.E7 Encodings: pMML, png, TeX See also: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23 … 3: 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 ) . … 4: 23.16 Graphics … ►See Figures 23.16.1–23.16.3 for the modular functions λ , J , and η . … ► ►► Figure 23.16.1: Modular functions λ ( i y ) , J ( i y ) , η ( i y ) for 0 ≤ y ≤ 3 . … Magnify … 5: 23.18 Modular Transformations … ► Klein’s Complete Invariant ► 23.18.4 J ( 𝒜 τ ) = J ( τ ) . ⓘ Symbols: J ( τ ) : Klein’s complete invariant, τ : complex variable and 𝒜 : bilinear transformation Referenced by: §23.18 Permalink: http://dlmf.nist.gov/23.18.E4 Encodings: pMML, png, TeX See also: Annotations for §23.18, §23.18 and Ch.23 … 6: 23.17 Elementary Properties … ► 23.17.5 1728 J ( τ ) = q − 2 + 744 + 1 96884 q 2 + 214 93760 q 4 + ⋯ , ⓘ Symbols: J ( τ ) : Klein’s complete invariant, q : nome and τ : complex variable Referenced by: §23.17(ii) Permalink: http://dlmf.nist.gov/23.17.E5 Encodings: pMML, png, TeX See also: Annotations for §23.17(ii), §23.17 and Ch.23 …