[Math] Accumulation points in complex analysis.

complex-analysis

I'm following the book Complex Variables and Applications by Churchill-Brown and I'm having trouble with a problem. It says the following:

  • Determine all the accumulation points of $z_{n} = (-1)^n (1+i)\frac{n-1}{n}$

The book gives the answers which are $\pm (1+i)$, and I think I understand the concept of an accumulation point. Basically, it's a point $z_0$ such that, if you take a deleted neighborhood around it, at least one point within the neighborhood lies in the original set. What I have trouble with is the fact of taking the original set $z_n$ and using the definition of the deleted neighborhood $0 < |z – z_0 | < \epsilon $, and finding those accumulation points.

At this point in the book there are no such concepts as sequences, limits or anything related to that, just basic definitions of neighborhoods and sets in the complex plane.

Best Answer

Note that $z_{2n}\to1+i$ and $z_{2n+1}\to-1-i$, and $z_{2n}\ne 1+i$ and $z_{2n+1}\ne -1-i$, so $\pm(1+i)$ are accumulation points.

Now

$$|z_{2n}-(1+i)|=\frac{1}{2n}|1+i|=\frac{\sqrt 2}{2n}$$ $$|z_{2n+1}-(-1-i)|=\frac{1}{2n+1}|1+i|=\frac{\sqrt 2}{2n+1}$$

So you might see that:

  1. Inside any deleted neigborhood of $\pm(1+i)$ there are an infinity of terms (all terms for which$\frac{\sqrt 2}{2n}<\epsilon$, or $\frac{\sqrt 2}{2n+1}<\epsilon$ respectively, so $\pm(1+i)$ are accumulation points
  2. Outside any union of two neigborhoods of $\pm(1+i)$ there are only a finite number of terms of the sequence.

Now take $w\in\mathbb C$, $w\ne\pm(1+i)$ and prove that it is not an accumulation point. Take a neighborhood $U$ of $w$ small enough so it doesn't contain any of the two points $\pm(1+i)$. If U does not contain any point of $z_n$, then we are done. Otherwise, take a deleted neigborhood $V$ of radius less than $\min_{z_n\in U, z_n\ne w}|w-z_n|$. It is easy to see that the deleted neigborhood does not contain any elements of $z_n$, which was to be proved.

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