描述J-inv-modulus.jpeg	Klein's J-invariant, modulus portrait (600x600 pixels)
日期	2005年5月21日, 21:21 (原始上传日期)
来源	本檔案是由Keyi從en.wikipedia轉移到維基共享資源。
作者	en:user:Linas
授权 (二次使用本文件)	​en 用户 Linas，本作品著作权人，特此采用以下许可协议发表本作品： 已授权您依据自由软件基金会发行的无固定段落及封面封底文字（Invariant Sections, Front-Cover Texts, and Back-Cover Texts）的GNU自由文件许可协议1.2版或任意后续版本的条款，复制、传播和／或修改本文件。该协议的副本请见“GNU Free Documentation License”。http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

原始上传日志

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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已授权您依据自由软件基金会发行的无固定段落及封面封底文字（Invariant Sections, Front-Cover Texts, and Back-Cover Texts）的GNU自由文件许可协议1.2版或任意后续版本的条款，复制、传播和／或修改本文件。该协议的副本请见“GNU Free Documentation License”。 受免責聲明的約束。http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

Relevant Links

• Weierstrass elliptic functions
• Eisenstein series
• Q-series