Playing with complex numbers

I’m just keeping my mind busy, ordering some thoughts and testing my memory.

Almost 20 years ago I was lucky to have a math teacher willing to teach our class ‘all about’ complex numbers. At that time there were some changes to the curriculum and he could have skipped that or chosen something else. I don’t remember what the alternative could have been, probably stochastics (I just couldn’t resist) ;-)

One part of it was analytical transformations. Of course it wasn’t called that way back then at school and we weren’t told what criteria complex valued functions have to meet to be called analytical either, because that needed knowledge about derivatives, possibly integration and a whole lot about a guy named Cauchy. And of course calculating derivatives and integrals was part of next year’s math curriculum.

But we were taught quite a lot about

And of course there’s the famous:

BTW, complex math is the source of the world famous Mandelbrot set.

Here’s what I’ve found in the deep chasms of my mind today:

This little formula acts as a mirror using the dashed circle of radius 1 as the ‘axis’. Funny things happen.

If I’m not mistaken this transformation here is actually not analytical. If it were it should reproduce the 90° angles of the square perfectly. One textbook application of analytical transformations is calculating equipotential lines and electric fields in unpleasant geometries. As everybody should know field lines and equipotential lines are orthogonal. This fact is preserved if transforming geometries of electrodes into a more pleasant shape, meaning that calculation becomes easy or even trivial.

And the last one:

Here’s the files that were used to create the images. Some Inkscape post processing is necessary.

circle_mirror.pl , makeplot_circle_mirror.txt

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