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I just discovered that complex functions exist, and that you need four dimensions to represent them (i.e. three space dimensions and one represented by colours => Riemman’s surfaces). I was wondering if there is any software or programme that can help me visualise an arbitrary complex function that I want to input. I would be really grateful if someone could help me with this. I love seeing functions alive, especially in four dimensions :)

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  • I suspect any such software will output unreadable results. To begin with, the four variables correspond to the real and imaginary parts of the argument and value of the function, only three of these can be represented by numeric axes and the fourth one will have to be colour but the lack of symmetry on the meaning make this already dubious. Also, you'd end up with a solid coloured in a gradient throughout (not a nice sheet you can visualise). – William M. May 06 '22 at 20:31
  • What seems sensible to do is to think of such a complex function as a vector field from the plane to itself, and plot it as a vector field. (I.e. you plot the arrow $f(z)$ emerging from $z.$) – William M. May 06 '22 at 20:33
  • @carefree that function plots $|f(z)|$ and colours it as a function of the principal argument of $f(z)$, which is not quite what OP wants. – William M. May 06 '22 at 20:49
  • I think your comment.is good because it shows what people actually do to visualise complex functions. I think either the vector field approach of the Mathematica approach are the most sensical. – William M. May 06 '22 at 20:53
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    In 3D you may plot the real part, the imaginary part, the modulus (and the phase) using for example Mupad (integrated in matlab) here and here. You may too plot the image of a line of the complex plane like zeta zeros. Domain coloring is another possibility. Concerning 4D you may try this last animation. – Raymond Manzoni May 06 '22 at 21:26

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