描述J-inv-modulus.jpeg	Klein's J-invariant, modulus portrait (600x600 pixels)
日期	2005年5月21日, 21:21 (原始上傳日期)
來源	Transferred from en.wikipedia to Commons by Keyi.
作者	en:user:Linas
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All following user names refer to en.wikipedia.

This image shows the modulus |j| of the J-invariant ${\displaystyle j=g_{2}^{3}/\Delta }$ as a function of the square of the nome ${\displaystyle q=\exp(i\pi \tau )}$ on the unit disk |q| < 1. That is, ${\displaystyle \pi \tau }$ runs from 0 to ${\displaystyle 2\pi }$ along the edge of the disk. Black indicates regions where the modulus is near zero, green where the modulus is about one, and red where the modulus is greater than ten. The color scale is logarithmic. Inside of every black region is a zero; note the zeros are cubic.

The small black dot on the far right, in the large red cardioid, is a numerical artifact.

The fractal self-similarity of this function is that of the modular group; note that this function is a modular form. Every modular function will have this general kind of self-similarity. In this sense, this particular image clearly illustrates the tesselation of the q-disk by the modular group.

• See also Image:J-inv-poincare.jpeg for this J-invariant on the Poincaré disk.
• See also Image:J-inv-real.jpeg for the real part.
• See also Image:J-inv-phase.jpeg for the phase.

It, and other related images, can be seen at http://www.linas.org/art-gallery/numberetic/numberetic.html

Created by Linas Vepstas User:Linas <[email protected]> on 21 May 2005 using custom software written entirely by Linas Vepstas.

Released under the Gnu Free Documentation License (GFDL) by Linas Vepstas.

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• Weierstrass elliptic functions
• Eisenstein series
• Q-series