# Complex analysis limit points question

• saraaaahhhhhh
In summary, according to the author, limit points in Complex Analysis are conceptually identical to limit points in Real Analysis, and b) would be the set of all pts Imz=0. For c), 1+i (all go to infinity) would be the limit point, while 1, 0, p/m, and iq/n would be the accumulation points. For a), there are no limit points found.
saraaaahhhhhh

## Homework Statement

Find the limit points of the set of all points z such that:
a.) $$z=1+(-1)^{n}\frac{n}{n+1}$$ (n=1, 2, ...)
b.) $$z=\frac{1}{m}+\frac{i}{n}$$ (m, n=+/-1, +/-2, ...)
c.) $$z=\frac{p}{m}+i\frac{q}{n}$$ (m, n, p, q=+/1, +/-2 ...)
d.) $$|z|<1$$

None.

## The Attempt at a Solution

I'm unsure on a.
I'm also unsure on b. I think it's just a bunch of points starting at the line 1 above the real axis and going down. But not totally filled in, so I'd think there'd be no limit points. But then again maybe 0 is a limit point?
c.) I think it's the set of all pts Imz=0
d.) I think it's the set of all pts |z|=1

This is problem 1 on page 29 in Introductory Complex Analysis by Silverman...it's on google books.

What's your definition of limit points? All points in the set plus all accumulation points (where accumulation points are defined to be points where all neighborhoods containing them intersect with the original set), or just the latter? Based on your answers, I'm guessing the latter.

If it's just the latter, then your d) is correct. Did you take real analysis? If you did, limit points in C are conceptually identical to limit points in R.

For b), it's a bit tricky. First, let both m and n go to infinity. This will be a limit point, right? After you've done that, fix m (for example, take m = 3), and let n go to infinity. This will also be a limit point. Same thing if you fix n and let m vary. Once you're familiar with this method, you can do c), too.

So b would be 1/m, i/n, and 0?
And c would be 1+i (all go to infinity), 1, 0, p/m, and iq/n?

What about a? Are there no limit points? It doesn't seem to converge anywhere. Except maybe at 1 and 2, when n goes to infinity?

## What is complex analysis?

Complex analysis is a branch of mathematics that deals with complex numbers and functions. It is used in various fields of science, including physics, engineering, and economics.

## What are limit points in complex analysis?

In complex analysis, a limit point is a point on the boundary of a set where the function approaches infinity. It is also known as a point of accumulation.

## Why are limit points important in complex analysis?

Limit points play a crucial role in the study of complex functions, as they help in determining the behavior of the function near the boundary of a set. They also help in understanding the convergence and divergence of a function.

## How do you find limit points in complex analysis?

The limit points of a function can be found by taking the limit of the function as it approaches the given point. If the limit is finite, then the point is a limit point. If the limit is infinite, then the point is not a limit point.

## Can limit points have complex coordinates?

Yes, limit points can have complex coordinates in complex analysis. This is because complex numbers have both a real and imaginary part, allowing for more complex behavior near the boundary of a set.

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