Pictures of some generalized analytical continuations

[Caleb]

Now hold on a minute.
 If I am not mistaken you took micks function and provided an alternative series expansion that converges in a larger domain.

But now you use the alternative series expansion and plug in 1/2 instead of 2.

But if you took the original series for mick's function it converges better if you pick 1/2 instead of 2.

So you got this one backwards ?

You only needed to replace 2 with 1/2 in micks original series, and that converges.

So if you do nothing or go backwards from your alternative series to the double sum , back to micks definition , then your problem is solved.

Then it looks like one of your simple "plug-ins".

Or did I make a mistake ?


The idea of using geometric series seems to work so often , apart from anyting that resembles ( with multiple sums , internal or external ) 
1+1+1+1+...

which means here that 1/2 and 2 work but 1 does not extend ?
Just an idea or the philosophy of the geometric series in my viewpoint.



regards

tommy1729

I'll edit the original post when I have better internet. For now, see this desmos graph: https://www.desmos.com/calculator/wulciifdwn. The main point is that when \(\alpha > 1 \), (like is the case in micks function), then all of the poles are INSIDE the unit circle. Thus, the regular meromorphic (not generalized analytical continuation), is able to pick up all of these poles. In the graph, I draw in the complex plane the location of where all the poles "should be," even for \( \alpha < 1 \). The problem is, that for \( \alpha < 1 \) you don't get to see any of these poles, since they are outside the natural boundary.

So, my suggestion was that one criteria for determining whether a generalized analytical continuation is good is to check whether it preverses where the poles are supposed to be. In particular, when we look at the slight variation of micks function where \( \alpha < 1 \), then we should be interested in checking whether the poles are at the locatoin they should be.