Need Help With Epsilon Neighbourhoods

[tanman24]

Hello,

I am a physics undergrad with no experience in analysis and I have not had to do any proofs before. I am taking a class in complex analysis as an elective. I have been doing well in the course so far and improving on my rigour in proofs and such. On to my question:

As epsilon-neighbourhoods were introduced, I understand it completely on an intuitional level, but am having trouble using it in proofs. Does anyone know any good resources with examples to use as I could not find anything.

Thanks for your time!

 

[tiny-tim]

hello tanman24! welcome to pf!

usually, you simply have to make a suitable choice for δ as a function of ε, eg δ = kε. δ = √ε …

sometimes you need to specify two conditions, eg δ = kε and δ < 1

if you do a search on PF for "delta" and "epsilon", you should find plenty of examples ​

 

[brmath]

In complex analysis a neighborhood is a circle around the limit point. So say you are trying to prove that $f(z) \rightarrow L$ as $z \rightarrow z_0$. What you want to show formally is that if you pick a small distance $\epsilon$ between f(z) and L you can always find a circle around $z_0$ where |f(z) - L| < $\epsilon$ for every z inside the circle. The $\delta$ comes in because that is the radius of the circle.

Please note that although they never say this, the $\delta$ depends not only on $\epsilon$ but also on $z_0$.

Now if you are lucky, the f(z) will be something simple, so that you can find a simple function of $\epsilon$ in which you can express your delta, like $\delta = \epsilon/2$ , which might occur if f is a linear function; or $\delta = \sqrt \epsilon$ if you have a quadratic. Mostly one is not lucky and has to find another way. However, if you are taking problems out of a textbook, you'll likely fall into the lucky category.

You may not be finding material because you are looking under "complex analysis", whereas you want to look under elementary calculus. In that case your circle in the complex plane is replaced with an interval on the real line. There should be tons of examples of computing a $\delta$ in the calculus material on limits. But if you are still having trouble rounding up examples, ask and I will construct some for you. Formally (that is, notationwise), nothing changes in the complex plane. You replace the interval with a circle and proceed in pretty much the same way.

Underneath that benign notation something very serious has changed. Getting a limit to work coming in just from the left and right as you do on the real line, is quite different from getting it to work coming in from any arbitrary point in a circle. So for a limit to exist in the complex plane is a much stronger condition. There are plenty of examples of limits that converge nicely along the real axis and don't converge at all, or converge to something different along the imaginary axis.

The strength of that complex limit has immense consequences, which you will see as the course progresses.