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| Klein's [[J-invariant]], real part (600x600 pixels)
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| ===Detailed description===
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| This image shows the real part <math>\Re j</math> of the [[J-invariant]] <math>j=g_2^3/\Delta</math> as a function of the square of the [[nome (mathematics)|nome]] <math>q=\exp (i\pi\tau)</math> on the [[unit disk]] |''q''| < 1. That is, <math>\pi\tau</math> runs from 0 to <math>2\pi</math> along the edge of the disk. Black areas indicate regions where the real part is zero or negative; blue/green areas where the value is small and positive, yellow/red where it is large and positive. The diamond-shaped patterns in the red part are [[Moiré pattern]]s, and are an artifact of the pixelization of the image (the red-black strips are smaller than the size of a pixel; the color of the pixel is assigned according to the value of the function at the center of the pixel, rather than the average of values over the pixel).
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| 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.
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| The imaginary part <math>\Im j</math> looks roughly similar; the modulus <math>|j|=\sqrt{(\Re{j})^2+(\Im{j})^2}</math> is uniform in color, with the black strips filled in to match the colored areas. The modulus essentially looks like [[:Image:Q-euler.jpeg]] with the colors reversed.
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| Zeros are visible where-ever the three-pointed triangle tips come together. The j-invariant has a pole at every rational multiple of π on the circumference of the disk. The correct way to understand this image is to note that j takes on every possible value on the [[fundamental region]]. Each fundamental region takes the form of a hyperbolic triangle in this image, with one vertex of the triangle on the edge of the disk. Thus, the red regions deceivingly hint that they are centered on a pole; this is not the case, as the poles lie on the disk boundary. There is exactly one exception to this: there is one very tiny triangle (about two pixels in size), taking the shape of an oval, that is wrapped around the dead-center of the disk. One corner of that triangle is exactly at the center ''q''=0, with a pair of edges zipped together running from ''q''=0 to <math>q=-\exp(-\pi\sqrt{3})</math> (which is about -0.0043, which is why its not visible here). The third edge of this exceptional triangle circles the origin all the way around. This third edge is shared with the unique funny-looking tongue in this image, turning this seemingly two-sided tongue into a real triangle. Note that this implies that the j-function has a simple pole at the origin, although it is not visible in this image.
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| See also [[:Image:J-inv-phase.jpeg]] for the phase part.
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| It, and other related images, can be seen at http://www.linas.org/art-gallery/numberetic/numberetic.html
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| ===Source of Image===
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| Created by Linas Vepstas [[User:Linas]] <linas@linas.org> on 15 February 2005 using custom software written entirely by Linas Vepstas.
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| ===Copyright status===
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| Released under the Gnu Free Documentation License (GFDL) by Linas Vepstas.
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| {{GFDL-with-disclaimers|migration=relicense}}
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| ===Relevant Links===
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| * [[Weierstrass elliptic functions]]
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| * [[Eisenstein series]]
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| * [[Q-series]]
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| [[Category:Images of elliptic functions]]
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| {{badJPEG}}
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| {{Copy to Wikimedia Commons|bot=Svenbot|priority=true}}
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Nestor is the title my parents gave me but I don't like when individuals use my full title. Managing people is what I do in my day occupation. Playing croquet is something I will never give up. He currently lives in Arizona and his parents reside nearby.
my webpage; extended car warranty (similar internet page)