(02/28/2023, 07:18 PM)Caleb Wrote: This is an interesting analysis! I'll look over in detail later today-- but for now, I should mention that my intuition is that \(f(x)\) has a 'fake' natural boundary, but the other representation of \(f(x)\), where \(F(x) = \sum_{n=0}^\infty (-1)^n x^{2^n}\) has a 'real' natural boundary. The difference between a fake natural boundary and a real one is what I think drives the difference between the sums outside the natural boundary. I think if you try to do your analysis with the residues for \(F(x) = \sum_{n=0}^\infty (-1)^n (1/e)^{a^n}\) you should find that extra residues get picked up, and that they don't sum up to zero. Just a thought-- I'll definintely try this out myself later today and see what happens.
Oh you are definitely right. I don't think this works for \(g(x) = \sum_n (-1)^n x^{2^n}\). And for all wall of singularities situations. I think this function just happens to have a "removable" wall of singularities. Why I was so interested in this function. So, it's a bit of a tangent. But I'm sure, still somewhat relevant. I don't expect to find a formula for \(g\) from this, lmao. I'm just perplexed by this weird ass fucking relationship.
I made a post on MO, asking for references on similar phenomena of this \(f\) situation:
https://mathoverflow.net/questions/44183...nt-domains
Please upvote it; because it's the only way anyone of stature ever fucking reads it, lmao!
But this is definitely a bit of a tangent...
But still, this is the Kernel situation. When it goes to zero under the integral. Which can't hurt to know right?
Sincere regards.
