More math things! A few gentle tweets about one of my favorite theorems, Picard's Great Theorem.https://twitter.com/EmilyGorcenski/status/841670056199245827 …
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Replying to @EmilyGorcenski
You may know that dividing by zero leads a function to infinity. Picard's Great Theorem tells us that for some complex functions, we do more
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Replying to @EmilyGorcenski
Basically, some functions, like exp(1/z), the division by zero is so strong that in *any* punctured neighborhood around z=0...
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Replying to @EmilyGorcenski
the function assumes *every* complex value, except maybe a single one, in this case, zero. No matter how small a neighborhood you make.
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Replying to @EmilyGorcenski
So in other words, the theorem tells us sometimes we can divide by zero so hard, we basically make everything around us infinitely "dense."
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Replying to @EmilyGorcenski
So, as you approach the pole, you can find a sequence that converges to any given value, any way you want? Neat!
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Replying to @mikewasson
it's not a pole, it's an essential singularity. The two are distinct.
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poles can be removes by multiplying by (z-a)^n for big enough n. Essential singularities cannot.
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