Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - 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: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation (/showthread.php?tid=366) Pages: 1 2 RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - Base-Acid Tetration - 10/17/2009 hmm... i think the one at -3 would locally be a DOUBLE-logarithmic singularity, and the one at -4 is a triple-logarithmic singularity, etc. (if you take the log of 0 n times, you get a log^n singularity; if you exp^n a log^n singularity you get zero.) RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - bo198214 - 10/17/2009 (10/17/2009, 10:47 AM)andydude Wrote: It is well known that there is a logarithmic singularity at -2, which is a specific kind of essential singularity. A logarithmic singularity is not an essential singularity! All 3 types: removable singularity, pole and essential singularity, are isolated singularities, i.e. the function is holomorphic in vicinity except at that singularity. This is not the case for the logarithm at 0. 0 is a branchpoint of the logarithm and of roots. Quote:It is also known that there are essential singularities at -3, -4, etc, but exactly what kind of singularities these are is not well known. Jay would call them doubly, trice, etc logarithmic. These all are branchpoints not isolated singularities. RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - andydude - 10/17/2009 (10/17/2009, 04:37 PM)bo198214 Wrote: A logarithmic singularity is not an essential singularity! Sorry, I thought a logarithmic singularity was both a singularity and a branchpoint. RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - Base-Acid Tetration - 10/17/2009 Not very rigorous argument: The region around -2 is locally a logarithmic branchpoint. At -2 every branch of tetration, like those of the logarithm, falls to an infinite value. since tet(-3)= log(tet(-2)) and logarithm of infinity, whatever infinity it is, is a kind of essential infinity, so every branch of tetration has these singularities we can repeat this process to get more than one singularities for every branch of tetration, which an entire pentation must avoid, so any entire pentation is trivial, a constant function pen(z) = fixed point of tet(z). so pentations that we like won't be entire. RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - andydude - 10/23/2009 Ok, I think I know what you were talking about now. The primary branchpoints of pentation are going to be where you see the logarithmic singularities in the picture below. These are approximately at $-1.68187 \pm 1.33724 I$ which is also approximately ${}^{\infty}e - 2$. It is not exact, because the region between -1 and 0 is only approximately (x+1). In other words, if L is a branchpoint of the superlogarithm, then $slog(slog(L))$ is going to be branchpoint of pentation. I also plotted two branch systems: A, from branchpoints to imaginary infinities D, from branchpoints to negative infinity The function being plotted is actually: $f(z) = \begin{cases} \text{slog}(f(z + 1)) & \text{if } Re(z) < 0 \\ {}^{z}e & \text{if } 0 \le Re(z) \le 1 \\ \text{sexp}(f(z - 1)) & \text{if } Re(z) > 1 \end{cases}$ Andrew Robbins RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - andydude - 10/23/2009 I also remember someone asking about fixedpoints of tetration, so I graphed some complex plots of $e^z - z$ and ${}^{z}e - z$ for comparison. It looks as though there are fixedpoints of tetration at: $z = -1.85$ $z = 0.8 \pm 1.5 i$ $z = 3 \pm 0.5 i$ and more... RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - Base-Acid Tetration - 10/23/2009 Is it me, or will pentation indeed look a lot like tetration in the complex plane? RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - andydude - 10/24/2009 (10/23/2009, 09:27 PM)Base-Acid Tetration Wrote: Is it me, or will pentation indeed look a lot like tetration in the complex plane?Well, if by "look like" you mean it will have 2 branchcuts ($-1.6 \pm 1.33i$) instead of 1 branchcut (-2), and 4 primary fixedpoints instead of 3 primary fixedpoints, then yes. Below is a complex plot of $\text{pexp}(z) - z$, and even though there is a bit of double-vision, it is clear that there are 4 fixedpoints near the origin. RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - Base-Acid Tetration - 10/24/2009 (10/24/2009, 12:26 AM)andydude Wrote: Well, if by "look like" you mean it will have 2 branchcuts ($-1.6 \pm 1.33i$) instead of 1 branchcut (-2), and 4 primary fixedpoints instead of 3 primary fixedpoints, then yes. I mean the way they both decay to a conjugate of values at large imaginary parts and positive real part. RE: Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation - andydude - 10/24/2009 (10/24/2009, 12:31 AM)Base-Acid Tetration Wrote: I mean the way they both decay to a conjugate of values at large imaginary parts and positive real part. Yes, it appears that $\lim_{x\to\infty} e{\uparrow}^3 (i x) = 0.7648667180537022 + 1.5298974233945777i$ (obtained with $\text{sexp}^{\infty}(1+1.5i)$) $\lim_{x\to\infty} e{\uparrow}^3 (-i x) = 0.7648667180537022 - 1.5298974233945777i$ (obtained with $\text{sexp}^{\infty}(1-1.5i)$) $\lim_{x\to\infty} e{\uparrow}^3 (-x) = -1.85035662730682$ (obtained with $\text{slog}^{\infty}(-1.5)$) do you mean negative real part?