If you want some visuals, here is something I made a few months ago: https://math.berkeley.edu/~kmill/math54fa16/roots/roots.html https://math.berkeley.edu/~kmi

If you want some visuals, here is something I made a few months ago: https://math.berkeley.edu/~kmill/math54fa16/roots/roots.html https://math.berkeley.edu/~kmill/math54fa16/roots/roots.html (it accepts polynomials in any reasonable format, factored or not, and gives both the domain coloring and the circle image). I thought I had implemented a visualization for Gauss's original proof, but I only did that for the reverse problem of giving the roots (and poles) and seeing the zero curves for the real and imaginary parts: http://www.kylem.net/stuff/homsim/homsim2.html http://www.kylem.net/stuff/homsim/homsim2.html (hit tab a few times). For dealing with non-polynomials, I made https://math.berkeley.edu/~kmill/tools/zgraph/zgraph.html https://math.berkeley.edu/~kmill/tools/zgraph/zgraph.html shortly after. It handles most complex functions, like polynomials, quotients, exponentiation, the exponential function, logarithms, and trigonometric functions (though possibly not inverse trigonometric functions). This gives the domain coloring along with the Riemann sphere. One day I'll add the ability to define functions so that one may, for instance, explore Moebius transformations. This second program works by compiling the input complex function into a shader program (which is output to the Javascript console, and it might be more interesting than the complex analysis itself to some people here).