DescriptionJ-inv-poincare.jpeg	Klein's J-invariant, modulus on Poincaré disk (600x600 pixels)
Date	22 May 2005, 01:30 (original upload date)
Source	Transferred from en.wikipedia to Commons by Keyi.
Author	en:user:Linas
Permission (Reusing this file)	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Subject to disclaimers. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original. This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. Subject to disclaimers.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

Linas at the English Wikipedia, the copyright holder of this work, hereby publishes it under the following license:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

The original description page is/was here. All following user names refer to en.wikipedia.

Klein's J-invariant, modulus on Poincaré disk (600x600 pixels)

This image shows the modulus |j| of the J-invariant ${\displaystyle j=g_{2}^{3}/\Delta }$ as a function on the Poincare disk. The Poincaré disk is a mapping of the upper half-plane to the unit disk by means of the Mobius transformation

${\displaystyle z={\frac {\tau -i}{\tau +i}}}$

where τ is the half-period ratio, that is, the coordinate on the upper half-plane. The left-right symmetry of this image corresponds to the ${\displaystyle \tau \to -1/\tau }$ symmetry of the J-invariant, as small values for the imaginary part of τ are mapped to z=−1 on the left hand side, while large values for the imaginary part of τ are mapped to z=+1 on the right side. The fundamental domain is a triangle spanning the two vertically oriented dots in the center, with the cusp at z=+1 on the right. The transformation ${\displaystyle \tau \to \tau +1}$ corresponds to walking the fundamental domain from dot to dot on the boundary of the largest red area on the right; the cusp z=+1 remains unchanged.

For a detailed description of other aspects of this picture, including the coloration, please refer to Image:J-inv-modulus.jpeg.

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

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

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Subject to disclaimers.
	You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
	This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. Subject to disclaimers.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

• Weierstrass elliptic functions
• Eisenstein series
• Q-series