User:Jan Homann/Mathematics

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Mathematics

Complex coloring.jpg


Polynomial Functions

Complex x hoch 3.jpg
Complex x hoch 5.jpg
Extrema complex x hoch 3.jpg
Extrema complex x hoch 5.jpg


Roots

Complex sqrt leaf1.jpg
Complex sqrt leaf2.jpg
Riemann surface sqrt.jpg
Complex cube root.jpg
Riemann surface cube root.jpg
Complex 4th root.jpg
Riemann surface 4th root.jpg


Rational Functions

Z over zminus1.jpg
Z2 over z2minus1.jpg
Z3 over z3minus1.jpg
Z4 over z4minus1.jpg
Z5 over z5minus1.jpg
Z7 over z7minus1.jpg
Extrema z over zminus1.jpg
Extrema z2 over z2minus1.jpg
Extrema z3 over z3minus1.jpg
Extrema z4 over z4minus1.jpg
Extrema z5 over z5minus1.jpg
Extrema z7 over z7minus1.jpg


Z3 over z5minus1.jpg
Z4 over z5minus1.jpg
Z5 over z5minus1.jpg
Z6 over z5minus1.jpg
Z7 over z5minus1.jpg


Z4 over z4minus1.jpg
Z4plus005 over z4minus1.jpg
Z4minus02i over z4minus1.jpg
Z4plus1 over z4minus1.jpg
Z4plus1 over z4minus005.jpg
Z4plus1 over z4.jpg
Extrema z4 over z4minus1.jpg
Extrema z4plus005 over z4minus1.jpg
Extrema z4minus02i over z4minus1.jpg
Extrema z4plus1 over z4minus1.jpg
Extrema z4plus1 over z4minus005.jpg
Extrema z4plus1 over z4.jpg


Exp, Gauss

Complex exp.jpg
Complex Exp minus z squared.jpg


Sin, Cos, Tan, Cot, Sec, Csc

Complex sin.jpg
Complex cos.jpg
Complex tan.jpg
Complex Cot.jpg
Complex Sec.jpg
Complex Csc.jpg


Sinh, Cosh, Tanh, Coth, Sech, Csch

Complex Sinh.jpg
Complex Cosh.jpg
Complex Tanh.jpg
Complex Coth.jpg
Complex Sech.jpg
Complex Csch.jpg


ArcSin, ArcCos, ArcTan, ArcCot, ArcSec, ArcCsc

Complex arcsin.jpg
Riemann surface arcsin.jpg
Complex arccos.jpg
Complex arctan.jpg
Complex ArcCot.jpg
Complex ArcSec.jpg
Complex ArcCsc.jpg


ArcSinh, ArcCosh, ArcTanh, ArcCoth, ArcSech, ArcCsch

Complex ArcSinh.jpg
Complex ArcCosh.jpg
Complex ArcTanh.jpg
Complex ArcCoth.jpg
Complex ArcSech.jpg
Complex ArcCsch.jpg


Log, ProductLog

Complex log.jpg
Riemann surface log.jpg
Product Log.jpg


PolyLog

Complex polylogminus3.jpg
Complex polylogminus2.jpg
Complex polylogminus1.jpg
Complex polylog0.jpg
Complex polylog1.jpg
Complex polylog2.jpg
Complex polylog3.jpg


Gamma, Reciprocal Gamma

Complex gamma.jpg
Complex Reciprocal Gamma.jpg


LogGamma, Polygammas

Complex LogGamma.jpg
Complex Polygamma 0.jpg
Complex Polygamma 1.jpg
Complex Polygamma 2.jpg
Complex Polygamma 3.jpg
Complex Polygamma 4.jpg


Riemann Siegel Theta (in 3 different magnifications), Riemann Siegel Z (in 2 different magnifications)

Riemann Siegel Theta 1.jpg
Riemann Siegel Theta 2.jpg
Riemann Siegel Theta 3.jpg
Riemann Siegel Z 1.jpg
Riemann Siegel Z 2.jpg


Riemann zeta, Dirichlet eta, Riemann Xi

Complex zeta.jpg
Complex Dirichlet eta function.jpg
Complex Riemann Xi.jpg


Eisenstein series

Eisenstein 4.jpg
Eisenstein 6.jpg
Eisenstein 8.jpg
Eisenstein 10.jpg
Eisenstein 12.jpg
Eisenstein 14.jpg


KleinInvariantJ, Modular lambda, Dedekind eta

KleinInvariantJ.jpg
Complex Modular Lambda.jpg
Dedekind Eta.jpg


Theta

Complex theta minus0point1times e i pi 0point1.jpg


Zeros of Riemann zeta function

Riemann Zeta.jpg
Color Funktion z.jpg




I used Mathematica 5.0 to generate the plots. I do not know how to add the corresponding Mathematica worksheet here. Therefore I took a screenshot of it for now, to make it possible for other people to reproduce the results. The color mapping is designed such that the phase of a complex number is always visible, even if the complex number approaches an absolute values of zero or infinity. You are welcome to generate further graphs. Please add the color map to the graph's description page! Please let me know about new graphs on my wikipedia user page Jan Homann!
There is a very nice version of the function for Mathematica 6.0 which you can directly download here [[1]]

Mathematica picture artefacts.jpg

Functions can show line-artefacts. To counter this, slightly expand the picture in the mathematica worksheet.


Mathematica coloring function.jpg