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Derivative of exp^[1/2] at the fixed point? - 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: Derivative of exp^[1/2] at the fixed point? (/showthread.php?tid=1043) Pages:
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Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/23/2015 Let exp^[1/2](x) be the half iterate of exp(x), generated by Kneser's Riemann mapping. Is the derivative defined of exp^[1/2](L) defined at the fixed point of L~=0.318132 + 1.33724i, where exp(L)=L? From an old 2010 post#3..post#8 http://math.eretrandre.org/tetrationforum/showthread.php?tid=544&pid=5400, we see that L is the closest singularity to the real axis for the half iterate, but it is a mild singularity, and that exp^[1/2](L)=L, so that exp^[1/2](L) is continuous at L. Is the derivative is also continuous at this singularity, and if so, then what is its value? If the derivative is continuous at the singularity, how many of the higher derivatives are also continuous? RE: Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/24/2015 (12/23/2015, 04:39 PM)sheldonison Wrote: Is the derivative is also continuous at this singularity, and if so, then what is its value? If the derivative is continuous at the singularity, how many of the higher derivatives are also continuous? In particular, there is a formal half iterate that is not real valued that can be developed at the fixed point L. And the formal half iterate begins with Numerical experiments suggest that the derivative of Knesser's real valued half iterate is continuous at L, and the first derivative at the singularity at L is RE: Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/25/2015 (12/24/2015, 03:25 AM)sheldonison Wrote: Numerical experiments suggest that the derivative of Knesser's real valued half iterate is continuous at L, and the first derivative at the singularity at L is I think the key equations in understanding this behavior are the slog, the sexp, the Schroeder equation at L, and the Abel equation generated from the Schroeder equation, and the theta mapping from the Abel equation to the slog. Kneser's exp^{0.5}: Here is the formal exp^{0.5} at L, generated from the formal Schroeder equation developed at the fixed point L Then the formal exp^{0.5} at L is exactly the same as: The next step is to show the Abel equation, developed from the Schroeder equation; where And Kneser's slog(z) developed from the Abel equation is: RE: Derivative of exp^[1/2] at the fixed point? - andydude - 12/27/2015 (12/24/2015, 03:25 AM)sheldonison Wrote: And the formal half iterate begins with Ok, so I replaced y with 1/2 and log(L) with L in the regular iteration power series to get this: as expected it's the same power series. I wanted to highlight one of my findings in this paper (page 12) that is related but separate from this, which is a power series for Substituting in which I realize is a different base, but still interesting. Using a similar technique, we might be able to find a comparable power series for RE: Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/27/2015 (12/27/2015, 11:15 AM)andydude Wrote: Ok, so I replaced y with 1/2 and log(L) with L in the regular iteration power series to get this:Hey Andy, Thanks for your reply. Oops; I had a typo in my 2nd derivative which I fixed. I have a pari-gp program, that calculate the coefficients iteratively. Quote:I wanted to highlight one of my findings in this paper (page 12) that is related but separate from this, which is a power series for The parabolic case is hugely interesting. I usually work with iterating For the case at hand, RE: Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/29/2015 I wanted to post what I've found out so far; which is not complete, and explain why only four derivatives are defined for Kneser's half iterate at the fixed point of L. First, let S(z) be the Schroeder function, Then let The constant term Since So what we have for the individual theta(z) terms is I left out many details including all of details about how to derive RE: Derivative of exp^[1/2] at the fixed point? - andydude - 12/29/2015 (12/27/2015, 11:40 PM)sheldonison Wrote: Thanks for your reply. Oops; I had a typo in my 2nd derivative which I fixed. I have a pari-gp program, that calculate the coefficients iteratively. I don't think there was a typo, if you multiply the numerator and denominator by RE: Derivative of exp^[1/2] at the fixed point? - tommy1729 - 12/30/2015 Putting the issue in limit form (exp^[1/2](L+h i) - exp^[1/2](L-hi)) / h^n Where h is infinitesimal. Regards Tommy1729 RE: Derivative of exp^[1/2] at the fixed point? - sheldonison - 12/31/2015 I updated some of the equations in post#6 So then we have a conjectured equation for the real valued Kneser half iterate in terms of the formal half iterate which is as follows edit: fixed typos I think this is a complete form for the Kneser half iterate. Next I would like to calculate some of the RE: Derivative of exp^[1/2] at the fixed point? - tommy1729 - 12/31/2015 (12/29/2015, 10:25 PM)sheldonison Wrote: I wanted to post what I've found out so far; which is not complete, and explain why only four derivatives are defined for Kneser's half iterate at the fixed point of L. First, let S(z) be the Schroeder function, The 5 th derivative of of ?? No singularity ? Im sure you make sense , but it is not clear what you are doing to me. Regards Tommy1729 |