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I have the folowing problem at hand:

Find all singularities of $\sin\left(\frac{1}{\cos\left(\frac{1}{z}\right)}\right)$ and determine their type.

Now I believe the set of singularities are $\left\{ \frac{\pi}{2} + n \pi \colon n \in \mathbb{Z} \right\}$. But I find it hard to figure out what type they are from removable, poles, essential or not isolated at all. Can anyone help me?

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    $\begingroup$ Welcome to MSE! A general remark that is helpful in your case is the following: If $f:\mathbb C\to \mathbb C$ is holomorphic and not a polynomial (this is oftentimes called a transcendental function), and $g$ is holomorphic with a singular point $z_0$ that is not removable, then $f\circ g$ has in $z_0$ an essential singularity. So you can reduce yourself to studying the singularities of $\cos(1/z)$. $\endgroup$
    – sampleuser
    Commented Mar 30, 2020 at 11:00

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Singularities are the points $z$ such that $\cos (1/z)=0, $ which occurs when $1/z=\pi/2+n\pi$ so that you can write the values of $z$ now, where $n$ is any integer.

Second part: We have Tylor series expansion for $\sec z$ by which we can get the Laurent series expansion for $1/\cos (1/z)$ with infinitely many terms with negative powers of $z$. It is very well explained over here Singularities of $ {1}/{\cos(\frac{1}{z})}$

Therefore Sine function after composition has Laurent series expansion at $z=0$ with infinitely many terms having negative powers of $z.$ Hence the singularities are of essential type.

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  • $\begingroup$ This doesn't really answer the question which was regarding the types. $\endgroup$
    – sampleuser
    Commented Mar 30, 2020 at 11:23
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    $\begingroup$ Sorry I missed that part! $\endgroup$
    – user159888
    Commented Mar 30, 2020 at 12:02
  • $\begingroup$ No problem, think this now provides a nice answer to the OP‘s question. $\endgroup$
    – sampleuser
    Commented Mar 30, 2020 at 12:18
  • $\begingroup$ But isnt 0 a non isolated essential singularity $\endgroup$
    – Upstart
    Commented Jul 3, 2022 at 6:25

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