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+--- Thread: [MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n)) (/showthread.php?tid=1732)
RE: [MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n)) - Caleb - 04/07/2023
(04/06/2023, 07:54 PM)JmsNxn Wrote: Not to sound like a broken record, but:I appears most of this works out as I predicted, the Cauchy's integral theorem for fractional derivatives does give the desired result. I recently stubled upon this paper which proves that a neccesary AND sufficent condition for analytical continuation across arcs for a power series \( \sum f_n x^n \) is that \( f_n \) can be extended to an entire function of exponential type and that a_n is given by that cauchy integral theorem I mentioned before. I found this result from this MO answer: https://mathoverflow.net/a/369847/146528
*touches nose*
RE: [MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n)) - JmsNxn - 04/08/2023
Beautiful finds, Caleb.
This is definitely "in the neighborhood" of what I expected to happen. But it's still awesome to see the rigor in action. This isn't my topic of interest that I like to work in; but it's still super cool. Keep up the good work
Regards, James