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Could be tetration if this integral converges - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Could be tetration if this integral converges (/showthread.php?tid=847) |
RE: Could be tetration if this integral converges - JmsNxn - 05/07/2014 (05/07/2014, 03:25 AM)mike3 Wrote: Yes that should read As for your question on the tetration integral not working I hsould mention with I'm really stumped on applying this to tetration at the moment. But I feel like theres got to be a way using fractional calculus, I'm just missing it. RE: Could be tetration if this integral converges - mike3 - 05/11/2014 (05/07/2014, 03:18 PM)JmsNxn Wrote:(05/07/2014, 03:25 AM)mike3 Wrote: I'm curious here: what kind of strategies are available for evaluating that integral for RE: Could be tetration if this integral converges - tommy1729 - 05/11/2014 (05/03/2014, 01:19 AM)mike3 Wrote: JmsNxn, I totally agree. Although I added a minus sign to it, which I assume was a typo. So lets think about Since the Taylor coefficients of ( many remainder theorems for Taylor series imply this ) This means the main behaviour of this This implies that Therefore the integral diverges. Even if we consider taking the limit of x going to +oo as the limit of the sequence x_i with regards tommy1729 RE: Could be tetration if this integral converges - JmsNxn - 05/11/2014 If When F has singularities we see we pull on a second balancing function For simple functions like As in, if where I haven't looked into much of how these balancing functions behave. I'm more familiar with just working with entire And on a different note. I've successfully shown that, if where it satisfies the composition rule and interpolates the iterated continuum sum at natural values. Pcha!! Holomorphic in z and s. Pcha! RE: Could be tetration if this integral converges - JmsNxn - 05/11/2014 (05/11/2014, 04:26 PM)tommy1729 Wrote: I totally agree. Although I added a minus sign to it, which I assume was a typo. yes yes, I'm quite aware it diverges. That's only another trick we need to come up with to handle that. I have a few but I need to look deeper into the laplace transform. RE: Could be tetration if this integral converges - tommy1729 - 05/11/2014 (05/11/2014, 04:30 PM)JmsNxn Wrote:(05/11/2014, 04:26 PM)tommy1729 Wrote: I totally agree. Although I added a minus sign to it, which I assume was a typo. I assumed you were aware of it. But some readers might not have been convinced. With that in the back of my mind, I felt the neccessity to reply. Its pretty hard to combine the properties of convergeance and the functional equation with fractional calculus and integral transforms ... or so it seems. Maybe a bit of topic but finding an approximation to where (And the factorial is computed with the gamma function ofcourse ) with approximation I mean that they have the same " growth rate ". regards tommy1729 RE: Could be tetration if this integral converges - mike3 - 05/11/2014 (05/11/2014, 04:29 PM)JmsNxn Wrote: If Well, I guess then these formulas aren't of much use for continuum-summing tetration, since it is most definitely not bounded with the bound . In fact it is unbounded on the right half-plane (where it behaves chaotically) and has branch point singularities (which are neither poles nor essential singularities) on the left half-plane (these are logarithmic, double-logarithmic, triple-logarithmic, etc. in that order at I'm curious: how did you get that first formula? Is it possible to get a similar formula for and and RE: Could be tetration if this integral converges - JmsNxn - 05/12/2014 (05/11/2014, 11:26 PM)mike3 Wrote: I'm curious: how did you get that first formula? Is it possible to get a similar formula for I'm indisposed at the moment but the formula is derived in the paper I posted. Its a very brief proof and follows from cauchy's residue theorem and a meromorphic representation of the Gamma function. I could write some of it out, but it wouldn't be completely formal and might not leave you convinced ![]() RE: Could be tetration if this integral converges - mike3 - 05/12/2014 (05/12/2014, 01:44 AM)JmsNxn Wrote:(05/11/2014, 11:26 PM)mike3 Wrote: I'm curious: how did you get that first formula? Is it possible to get a similar formula for Contour integration, right? RE: Could be tetration if this integral converges - JmsNxn - 05/12/2014 (05/12/2014, 02:15 AM)mike3 Wrote:(05/12/2014, 01:44 AM)JmsNxn Wrote:(05/11/2014, 11:26 PM)mike3 Wrote: I'm curious: how did you get that first formula? Is it possible to get a similar formula for Yeah. It's a very nifty contour integral. The gamma function is just so beautiful, I wish I could just kiss it. ![]() |