J-invariant
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In mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL defined on the upper half-plane of complex numbers. It is the unique such function
Klein's j-invariant in the complex plane
Real part of the j-invariant as a function of the square of the nome on the unit disk
Phase of the j-invariant as a function of the square of the nome on the unit disk
Dedekind eta function
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In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part
J-invariant
…natural boundary at the real line. The given functions are the modular discriminant Δ ( τ ) = g 2 ( τ ) 3 − 27 g 3 ( τ ) 2 = ( 2 π ) 12 η ( τ ) 24 , Dedekind eta function η ( τ ) , and modular invariants, g 2 ( τ ) = 60 G 4 ( τ ) = 60 ∑ ( m , n ) ≠ ( 0 , 0 ) ( m + n τ ) − 4 g 3 ( τ ) = 140 G 6 ( τ ) = 140 ∑ ( m , n…
Dedekind η-function in the upper half-plane
Modulus of Euler phi on the unit disc, colored so that black = 0, red = 4
The real part of the modular discriminant as a function of q.