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Klein's J-invariant, modulus on Poincaré disk (600x600 pixels)
Detailed description
This image shows the modulus |j| of the J-invariant 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
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 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 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.
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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
Relevant Links
Weierstrass elliptic functions
Eisenstein series
Q-series
Captions
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{{Information |Description=Klein's J-invariant, modulus on Poincaré disk (600x600 pixels) |Source= Transferred from [http://en.wikipedia.org en.wikipedia]; transferred to Commons by User:Keyi. |Date=01:30, 22 May 2005 (original upload date) |Author=[
File usage
The following pages on the English Wikipedia use this file (pages on other projects are not listed):