Super-logarithm on the imaginary line
#1
I just found a way to calculate slog on the imaginary axis!
It depends very much on Jay's observation that slog is imaginary-periodic.

Let . S is periodic with period , because . Since S is periodic, we can use Fourier series to represent it. Let , then . The Taylor series coefficients of R will then be the Fourier series coefficients of S. In terms of the super-logarithm, . This means the Fourier series coefficients of are the Taylor series coefficients of which we already know. In other words, , so:


The nice thing about this is that it seems to bypass the radius of convergence problem near z=i since its a Fourier series and not a Taylor series. Is this right?

I've included a plot with multiple approximations, which seem to converge much faster than doing analytic continuation section-by-section. The top line is the imaginary part, and the bottom line is the real part, and the "y" axis is :

PDF version

[Image: superlog-imaginary.png]

Andrew Robbins
Reply
#2
Actually, if you think about it, we're still limited by the same radius of convergence, because it's the same series.

HOWEVER, because we're plugging e^z into the series, rather than z, we're limited by abs(e^z) < 1.37445, not abs(z)<1.37445.

This essentially means that for all complex values with real part less than 0.31813 (real part of primary fixed point), the series will converge.

So if you plug e^(iz) into the series, and use only real z, then (because the imaginary part is 0, and hence the real part of iz is 0) you will always be inside the radius of convergence. Effectively, you're calculating points on the unit circle with the original series.
~ Jay Daniel Fox
Reply
#3
By the way, now that I see it written down, I see that I made a simple but important mistake a few days ago, by forgetting the i in the exponents for the exponential form of the fourier series. I knew that we had real exponents for slog, but didn't quite link it up with the fact that we're periodic in the imaginary direction.
~ Jay Daniel Fox
Reply
#4
Right, since the real part of all points in the "backbone" of the slog are less than the logarithm of the radius of convergence, the exponential of them is within the radius of convergence (of the series expansion about z=0).

Andrew Robbins
Reply


Possibly Related Threads…
Thread Author Replies Views Last Post
Question Convergent Complex Tetration Bases With the Most and Least Imaginary Parts Catullus 0 254 07/10/2022, 06:22 AM
Last Post: Catullus
Question Derivative of the Tetration Logarithm Catullus 1 377 07/03/2022, 07:23 AM
Last Post: JmsNxn
  Imaginary iterates of exponentiation jaydfox 9 13,280 07/01/2022, 09:09 PM
Last Post: JmsNxn
Question Iterated Hyperbolic Sine and Iterated Natural Logarithm Catullus 2 571 06/11/2022, 11:58 AM
Last Post: tommy1729
  Is bugs or features for fatou.gp super-logarithm? Ember Edison 10 17,538 08/07/2019, 02:44 AM
Last Post: Ember Edison
  Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 18,477 06/10/2019, 04:29 AM
Last Post: Ember Edison
  Inverse super-composition Xorter 11 27,506 05/26/2018, 12:00 AM
Last Post: Xorter
  The super 0th root and a new rule of tetration? Xorter 4 10,119 11/29/2017, 11:53 AM
Last Post: Xorter
  Is the straight line the shortest, really? Xorter 0 3,186 05/23/2017, 04:40 PM
Last Post: Xorter
  Solving tetration using differintegrals and super-roots JmsNxn 0 4,010 08/22/2016, 10:07 PM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)