Laurent series z0=infinity

Hi there. I have to find the Laurent series for

So what I did was calling , I don't know if this is right.

Then:

And then back to z:

I have to make the series for |z|>2, but I see that it gives the same as when I centered the series in zero, so I think I'm doing something wrong.

Re: Laurent series z0=infinity

We have , expansion valid for or equivalently for .

Re: Laurent series z0=infinity

Yes, thats the other part for the Laurent series, but is it centered at ? because I've made the expansion centered at zero, and it gives exactly the same it gives when centered at infinity.

Re: Laurent series z0=infinity

Quote:

Originally Posted by

**Ulysses** Yes, thats the other part for the Laurent series, but is it centered at

?

Of course, is a neighborhood of .

Quote:

because I've made the expansion centered at zero, and it gives exactly the same it gives when centered at infinity.

What? I see different expansions.

Re: Laurent series z0=infinity

What's the difference for I just don't see it. I've proceeded exactly as you did when finding the Laurent series for this same function centered at zero, and I get the same series that we've found here, with exactly the same expressions for the modulus.

If I take then I get:

That's the Taylor part, the other gives the same that you found.

Re: Laurent series z0=infinity

Let us see, you have __completed__ the Laurent expansion valid for that is, __centered at__ which is the same than the Taylor expansion (there is no principal part). On the other hand we have __completed__ the Laurent expansion valid for that is , in another region __centered at__ . Right? Then, in the second case __we also say__ that we have the expansion centered at because is a neighborhood of .

Re: Laurent series z0=infinity

Thank you Fernando :D I was reading some of your works by the way. That poly-dimensional time that you've proposed, and the movement of movements it sounds like a really interesting thing. It's not something that we should discuss right here, I can't discuss it anyway, but I wanted to mention it :P

Bye bye! see you around.