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Definition 1: The point $z_0$ is called a singular point of a function $f$ if $f$ fails to be analytic at $z_0$ but is analytic at some point in every neighborhood of $z_0$

Definition 2: A singular point $z_0$ is said to be isolated if, in addition,there is a deleted neighborhood of $z_0$ throughout which $f$ is analytic

Example: The function $$\frac{1}{\sin(\frac{\pi}{z})}$$

has the singular points $z=0$ and $z=\frac{1}{n} (n=\pm 1,\pm2,\cdots)$, all lying on the segment of the real axis from $z=-1$ to $z=1$. $\underline{Each\; singular\; point \;except\; z=0 \;is\; isolated.}$

I'm confusing this underlined term, because i know clearly $z=0$ is not an isolated singular point, since every neighborhood of $0$ contains another singular points. This is ok.

But how the numbers of the form $\frac{1}{n}$ is isolated ? If i take, $n=1000000000000000000000$, then how we guarantee the neighborhood of this $\frac{1}{n}$ does not contains some remaining singular points ?

I'm sorry, this is very elementary for others, but i'm confusing

Any help is appreciated !

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1 Answer 1

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Given $n,$ the ball of radius $r=\min\{|\frac{1}{n}-\frac{1}{n+1}|,|\frac{1}{n-1}-\frac{1}{n}|\}$ centered at $\frac{1}{n}$ has no other singular points.

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  • $\begingroup$ Oh! .....I really appreciated! Thanks! $\endgroup$
    – user444830
    Jul 23, 2017 at 7:31

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