描述J-inv-modulus.jpeg	Klein's J-invariant, modulus portrait (600x600 pixels)
日期	2005年五月21日, 21:21 (原始上傳日期)
來源	本檔案是由Keyi從en.wikipedia轉移到維基共享資源。
作者	en:user:Linas
授權許可 (重用此檔案)	​en 使用者 Linas，此作品的版權所有人，決定用以下授權條款發佈本作品： 已授權您依據自由軟體基金會發行的無固定段落、封面文字和封底文字GNU自由文件授權條款1.2版或任意後續版本，對本檔進行複製、傳播和／或修改。該協議的副本列在GNU自由文件授權條款中。http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

原始上傳日誌

All following user names refer to en.wikipedia.

Detailed description

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

Source of Image

Created by Linas Vepstas User:Linas <linas@linas.org> on 21 May 2005 using custom software written entirely by Linas Vepstas.

Copyright status

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

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Relevant Links

• Weierstrass elliptic functions
• Eisenstein series
• Q-series