Divergent Series and Analytical Continuation (LONG post)

[tommy1729]

Okay, so take this post with a Grain of salt. I wanted to look deeper into this question. Because my suspicion is that we can say that sometimes, these are natural analytic continuations. I apologize if this is dumb. I'm just spit balling


I'd like to add my own two cents on:

\[
f(x) = \sum_{n=0}^\infty \frac{x^n}{1+x^n} \frac{1}{2^n}\\
\]

First of all, the word I used to use for these functions is a function that maps \(\mathbb{C} \to \mathbb{C}\) upto a measure zero set in \(\mathbb{C}\). In this case, the measure zero set is the unit circle \(U\). So that \(f : \mathbb{C}/U \to \mathbb{C}\). There is absolutely zero way to analytically continue these functions, or refer to them as analytic continuations of each other. By standard literature; they are two functions which are holomorphic on disjointed domains of \(\mathbb{C}\).

But yes, ONE is more natural; obviously the one which is just plugging in the number. This is a fallacy that Cauchy called the Generality of Algebra. And it was a stark criticism of Euler. Who upon which, Euler would use the Generality of Algebra to get correct results. But he also got incorrect results.

...

Lets define

\[
f(x) = \sum_{n=0}^\infty \frac{x^n}{1+x^n} \frac{1}{2^n}\\
\]

and
 g(x)

\[
g(x) = \sum_{n=0}^\infty \frac{x^n}{1+x^n} \frac{1}{3^n}\\
\]

Now there probably exist entire functions A(z),B(z) such that


A(f(z)) = B(g(z))

The idea was that taylor series of f or g would still have the same natural boundary because their powers do.
(although the multiplicity changes)


This leads to the idea that if we accept the continuation by plug-in for f and g then

Q(z) as a taylor expansion in two variables (a(z),b(z))
where a(z) = f(z) , b(z) = g(z)

starts to make sense.


Also

 \[
t(x) = \sum_{n=0}^\infty \frac{x^n}{1+x^n} 2^n\\
\]

by using 1 + 2 + 4 + 8 + ... = -1

might make sense.

t(1) = 

 \[
t(1) = \sum_{n=0}^\infty \frac{1}{1+1} 2^n\\
\]

so t(1) = - 1/2.


But we must be careful , we do not know if using such ideas is self-consistant with eachother.


I think this is somewhat in the spirit of what Caleb wanted to do.


regards

tommy1729