241 words on Uni
There are two harmless cases – a removable singularity, which isn't a problem at all (think z/z – basically it's 1 but it's not defined at 0) and a pole (something like 1/z) which is rather harmless and easy to handle as well [cue the 'joke' about why they don't hijack planes in Poland –
I'm just a simple Pole in a complex plane].
The third case is called an essential singularity. It's very weird and chaotic. One fact about them is that close to the singularity the function will get arbitrarily close to any complex number. A bit hard to imagine how that should work. So I dug out this instructive picture seen above that I already used to explain this when teaching the topic in England. It shows contour lines of (the real part of) the function e1/z. It shows that the contour lines approach the singularity from different sides (many times in fact, if you look carefully), thus giving a feeling for how exactly you'll come close to any value.
The other image at the top of the post shows the same thing but in 3D rather than using contour lines, thanks to CPlot. Prettier, but probably not as instructive.
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