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I've seen many graphs of functions in the complex plane. For example the $\Psi_0(x)$. They look very nice and I wonder how may I graph them by hand. If and only if it is possible. Other ones I want to graph are $$\operatorname{Li}_s(z)$$. How do you know what colors to use in the graph?

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Before domain coloring came along, the customary way of presenting a function's complex plane behavior was through an "altitude chart", like the one here for the Faddeeva function, or this one for the Hankel function.

The book by Jahnke and Emde has more of these figures. Since you mentioned wanting to do them by hand, and coloring by hand might not prove feasible, you can start by sketching these contours.

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Given a function $f:\mathbb{C}\rightarrow\mathbb{C}$, you can color the complex plane by associating to each point a color based on the angle $f(z)/|f(z)|$ makes with the real axis. You can also change the brightness depending on the value of $|f(z)|$.

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