05/28/2017, 01:13 AM

Some number of years ago in 2011, I made some complex plane graphs of the inverse Schröder function super-imposed with Kneser's Chi-star function, but it turns out I never posted them online since I skipped right to the complex valued superfunction instead; http://math.eretrandre.org/tetrationforu...93#pid6193 Working directly with the Schröder and Chi-star function is probably more mathematically accessible. So I thought I would put together a post with some pretty pictures of Kneser's Chi-star function.

Let's start with the \( \Psi \) (or Schröder function) for exp(z) developed at the complex fixed point, L~=0.318132 + 1.33724i;

where exp(L)=L, and the multiplier \( \lambda \) at the fixed point is also L since \( \exp(L+\delta)\approx L+L\cdot\delta \Rightarrow\;\lambda=L \)

The defining equation for the Schröder function is \( \Psi(e^z) = \lambda\cdot \Psi(z) \). Remarkably, this function maps iterated exponentiation base e to multiplication by \( \lambda \). Of course \( \Psi(0) \) has what turns out to be a really complicated singularity.

Kneser's Chi-Star \( \chi(z) \) function maps the Schröder \( \Psi \) function of the real number line to get the Chi-star function. \( \chi(z)=\Psi\circ\Re\;\; \) This is the Schröder of the real number line. It is also probably the most natural first step in generating analytic real valued Tetration. And it is a very pretty function. From the definition of the \( \Psi \) then \( \chi(e^z)=\lambda\cdot\chi(z) \)

The first pretty picture I'm posting is the inverse Schröder \( \Psi^{-1}(z) \), covering the range of \( \pm 30 ... \pm 20i \), and superimposed with Kneser's Chi-star \( \chi(z) \) function. Of course, its difficult to grasp the details of this pretty picture. Lets start with the definition. Normally, one generates the Taylor series coefficients of the \( \Psi^{-1}(z) \) function iteratively; I usually work with numerical values of the coefficients. For reference, I include the closed form for the z^2 coefficient. It is an entire function.

\( \Psi^{-1}(\lambda z)=\exp(\Psi^{-1}(z))\;\;\;\Psi^{-1}(z)\approx \lambda+z+\frac{0.5z^2}{\lambda-1}+O(z^3)\;\;\; \) This is the function in the complex plane graph picture below.

[attachment=1255]

Here is another image that serves as a "key" to what the yellow segment superimposed on the picture above refers to. You can see that where the green curve approximately meets the red curve is approximately zero. Actually, the green segment ends at approximately -10^-78, and the red segment starts at +10^-78, and then the red segment continues until approximately 1-10^-78. Of course, there is a singularity at 0, so we can't extend the picture all the way to exactly zero! There is also a singularity at 1. And a singularity at e, and a singularity at \( e^{e} \) and a singularity at \( e^{e^e} \).... The Chi-star contour in the images covers roughly -infinity to Tet(6). Each time you iterate exp(x), you jump to a new curved segment that is L times larger than the segment containing x. Below, I show eight segments of the Chi-Star.

[attachment=1256]

If you could figure out how to map the various segments of the Chi-Star back to the real axis of the iterated Tetration function, then you would have a mathematical way to generate Tetration. The next step in that process is to generate a superfunction for exp base e by taking \( \Psi^{-1}(\lambda^z)\; \) But this superfunction is not real valued at the real axis due to the singularities at \( \Psi(\exp^{\circ n} (0)) \). But back to the Chi-star itself. Let's look at the singularity near zero. Can you guess where the singularity would extend to if we got even closer to zero? Here, I extended the section near zero to +/- 1/Tet(5.5). Notice that the red and green segments continue almost exactly together until perhaps surprisingly, they run almost coincident with the Tet(4..5) segment. But then they make a soft u-turn, near 1/Tet(5), and then join up with the Tet(5..6) segment!

[attachment=1257]

Black in these complex plane graphs corresponds to zero, so wherever it goes, it would be "black" in the complex plane graph of the \( \Psi^{-1} \) function. Eventually, you get the checkerboard pattern where the black and the white are right next to each other. But the soft turnaround occurs where the function is still black. The full image with the Chi-Star singularity extended as far as it can go is too busy to see the "star" in the Chi-Star. I included the lower-right corner from the first image, with the soft turnaround near 1/Tet(5), where the function is near a singularity for Tet(5). \( \ln(\ln(\frac{1}{\text{Tet(5)}}))=\text{Tet}(3)+\pi i \) This compares with Tet(3)=3814279, where \( \Psi(\text{Tet(3)}) \) is a true singularity but \( \Psi(\text{Tet}(3)+\pi i) \) is only near the singularity. So we get this turnaround where the true singularity continues on.

[attachment=1259]

Let's start with the \( \Psi \) (or Schröder function) for exp(z) developed at the complex fixed point, L~=0.318132 + 1.33724i;

where exp(L)=L, and the multiplier \( \lambda \) at the fixed point is also L since \( \exp(L+\delta)\approx L+L\cdot\delta \Rightarrow\;\lambda=L \)

The defining equation for the Schröder function is \( \Psi(e^z) = \lambda\cdot \Psi(z) \). Remarkably, this function maps iterated exponentiation base e to multiplication by \( \lambda \). Of course \( \Psi(0) \) has what turns out to be a really complicated singularity.

Kneser's Chi-Star \( \chi(z) \) function maps the Schröder \( \Psi \) function of the real number line to get the Chi-star function. \( \chi(z)=\Psi\circ\Re\;\; \) This is the Schröder of the real number line. It is also probably the most natural first step in generating analytic real valued Tetration. And it is a very pretty function. From the definition of the \( \Psi \) then \( \chi(e^z)=\lambda\cdot\chi(z) \)

The first pretty picture I'm posting is the inverse Schröder \( \Psi^{-1}(z) \), covering the range of \( \pm 30 ... \pm 20i \), and superimposed with Kneser's Chi-star \( \chi(z) \) function. Of course, its difficult to grasp the details of this pretty picture. Lets start with the definition. Normally, one generates the Taylor series coefficients of the \( \Psi^{-1}(z) \) function iteratively; I usually work with numerical values of the coefficients. For reference, I include the closed form for the z^2 coefficient. It is an entire function.

\( \Psi^{-1}(\lambda z)=\exp(\Psi^{-1}(z))\;\;\;\Psi^{-1}(z)\approx \lambda+z+\frac{0.5z^2}{\lambda-1}+O(z^3)\;\;\; \) This is the function in the complex plane graph picture below.

[attachment=1255]

Here is another image that serves as a "key" to what the yellow segment superimposed on the picture above refers to. You can see that where the green curve approximately meets the red curve is approximately zero. Actually, the green segment ends at approximately -10^-78, and the red segment starts at +10^-78, and then the red segment continues until approximately 1-10^-78. Of course, there is a singularity at 0, so we can't extend the picture all the way to exactly zero! There is also a singularity at 1. And a singularity at e, and a singularity at \( e^{e} \) and a singularity at \( e^{e^e} \).... The Chi-star contour in the images covers roughly -infinity to Tet(6). Each time you iterate exp(x), you jump to a new curved segment that is L times larger than the segment containing x. Below, I show eight segments of the Chi-Star.

[attachment=1256]

If you could figure out how to map the various segments of the Chi-Star back to the real axis of the iterated Tetration function, then you would have a mathematical way to generate Tetration. The next step in that process is to generate a superfunction for exp base e by taking \( \Psi^{-1}(\lambda^z)\; \) But this superfunction is not real valued at the real axis due to the singularities at \( \Psi(\exp^{\circ n} (0)) \). But back to the Chi-star itself. Let's look at the singularity near zero. Can you guess where the singularity would extend to if we got even closer to zero? Here, I extended the section near zero to +/- 1/Tet(5.5). Notice that the red and green segments continue almost exactly together until perhaps surprisingly, they run almost coincident with the Tet(4..5) segment. But then they make a soft u-turn, near 1/Tet(5), and then join up with the Tet(5..6) segment!

[attachment=1257]

Black in these complex plane graphs corresponds to zero, so wherever it goes, it would be "black" in the complex plane graph of the \( \Psi^{-1} \) function. Eventually, you get the checkerboard pattern where the black and the white are right next to each other. But the soft turnaround occurs where the function is still black. The full image with the Chi-Star singularity extended as far as it can go is too busy to see the "star" in the Chi-Star. I included the lower-right corner from the first image, with the soft turnaround near 1/Tet(5), where the function is near a singularity for Tet(5). \( \ln(\ln(\frac{1}{\text{Tet(5)}}))=\text{Tet}(3)+\pi i \) This compares with Tet(3)=3814279, where \( \Psi(\text{Tet(3)}) \) is a true singularity but \( \Psi(\text{Tet}(3)+\pi i) \) is only near the singularity. So we get this turnaround where the true singularity continues on.

[attachment=1259]