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You might like the following animations: They show the
complex plots of sequences of rational functions whose poles accumulate. For
example, in the second figure you can see a sequence converging to a function
having the unit circle as its natural boundary. The limit function is
holomorphic in the open unit disc, but does not permit an analytic continuation
to any strictly larger domain in the complex plane. The first figure shows an
even more complicated situation. I don't know the locus of convergence.
And maybe it doesn't even matter, for that animation is just beautiful and I
quite like it just for its aesthetic value, irrespective of the mathematical
phenomena underneath it.
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Below, you find the complex plot of the generating function of the partition function. It is the key player in the Hardy-Ramanujan/Rademacher asymptotics for the problem to count partitions. As the functions above, it arises as a limit of rational functions with accumulating poles.
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