Think of the complex plane as having colors similar to those in a traditional color wheel. We put red at the complex number 1, with green and blue at the other two cube roots of unity as shown. Hues are interpolated, giving secondary and tertiary colors. A continuous blending would be possible, but here we show just twelve hues. Then we blend toward white at the center, toward black going outwards. Thus, each complex number has a color associated to it. |
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To visualize a complex-valued function in the plane we use what we call a domain coloring diagram: for each pixel in the domain of the function compute the color associated with that input value and use that color for that pixel.
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As a first example, here is a sixth degree polynomial with 4 simple zeroes and one double zero. You can spot the double zero because the colors cycle around twice when you make a circuit of that point in the domain. |
Next, observe the function f(z)=(z2-i)/(2z2+2i). This rational function has two zeroes (the white points) and two poles (the black ones). Each is simple. |
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Notice that in a neighborhood of the origin, the cyan color appears constant. This leads us to guess that the derivative of this function is zero there. We make a domain coloring diagram of f(z)-f(0) and observe a double zero: indeed the derivative is zero at the origin.
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Here is a domain coloring of Log(z). Observe that the complex number 1 is colored white, because the log of 1 is 0. The picture suggests that the derivative of Log(z) is 1 when z=1, because the color wheel is not distorted or turned (infinitesimally) near that point. |
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If you want to learn more about geometric approaches to complex function theory, I highly recommend Tristan Needham's wonderful new book Visual Complex Analysis. You can read portions of the text and get information about ordering from the web site linked to the title.
Link to homepage for Frank Farris.