Complex functions are ubiquitous objects in mathematics and science and since the early 19th century there has been exhaustive research by mathematicians all over the world, among them famous giants such as Riemann, Cauchy, Weierstrass or Gauss. Somehow they seem to appear everywhere: from number theory, where the Riemann Zeta function is arguably one of the most important functions (proving the famous Riemann hypothesis true will earn you a million dollars apart from everlasting fame) to fractals (you've probably seen a Mandelbrot set before) to deformation modelling and parametrization of surfaces in computer graphics.
However since the function graphs of complex-valued functions defined on (a subset of) the complex numbers live in general in Euclidean four-dimensional space they cannot be visualized traditionally. Therefore in order to plot them one has to come up with something different and the most popular choice is just using colors to encode the function values, leading to the well-known technique called "domain coloring". The following pictures give you an impression of how these plots may look like.
The function exp(1/z^2) has an essential singularity at zero
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The complex logarithm log(z)
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sin(z)/cos(z^2)
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The first 100 summands of the Riemann Zeta function series. Do all zeros lie on the vertical line with real part 0.5?
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cos(z) using a hand-drawn pencil sketch of a twig
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A meromorphic function: (z-1)(z+1)^2/((z+i)(z-i)^2)
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cos(log(z^2))
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tan(z)
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z^z
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References:
- K. Poelke, K. Polthier: Lifted Domain Coloring. Computer Graphics Forum, Volume 28, Number 3, June 2009 , pp. 735-742(8)
- K. Poelke, K. Polthier: Domain Coloring of Complex Functions: An Implementation-Oriented Introduction. Computer Graphics and Applications, IEEE. September-October 2012 , pp.6-13(8)
- M. Nieser, K. Poelke, K. Polthier: Automatic Generation of Riemann Surface Meshes. Advances in Geometric Modeling and Processing, Lecture Notes in Computer Science 6130, Springer Berlin/Heidelberg, pp 161-178, 2010.
- Domain coloring is also explained in the book Bilder der Mathematik, written by G. Glaeser and K. Polthier. You can find the relevant pages as an extract here (4.1MB).
Related Work:
- G. Semmler and E. Wegert have written a survey article on visualization techniques for complex function which appeared in the Notices of the Amer. Math. Soc., 58, (2011).
- H. Lundmark has a comprehensive web page on domain coloring and explains in detail what these plots reveal. See http://www.mai.liu.se/~halun/complex/
- F. Farris is said to have coined the term domain coloring. His colorings were made for a review for T.Needham's Visual Complex Analysis textbook and can be found under http://www.maa.org/pubs/amm_complements/complex.html
- Coloring algorithms for complex functions are also implemented in modern computer algebra systems such as Maple or Sage. There is even a Google Chrome plugin called Plomplex