# Can someone suggest me an non isolated singularity

Note by R G
4 years, 10 months ago

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Consider $$f(z) = \mathrm{cosec}(\tfrac{\pi}{z})$$. Then $$f(z)$$ has a pole at $$z=\tfrac{1}{n}$$ for any nonzero integer $$n$$, which makes the singularity at $$0$$ non-isolated.

- 4 years, 10 months ago

Thnxs dude can you think of one more function

- 4 years, 10 months ago

Generalize my previous example. Let $$f(z)$$ be any nonconstant entire function with an infinite number of zeros. Then the zeros are isolated, countable and unbounded. Then $$f(z^{-1})^{-1}$$ has a non-isolated singularity at $$z=0$$, since $$w^{-1}$$ is a pole for any zero $$w\neq0$$ of $$f(z)$$. For example, $$f(z) = e^z - 1$$ would do the trick.

- 4 years, 10 months ago