Hyperoperations Wiki - User contributions [en]

User:Catullus

2022-12-15T03:37:35Z

Catullus:

Hello. I am [https://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=310 Catullus]!

Euler number

2022-12-12T03:59:10Z

Catullus:

In our context, one can characterize the Euler number e in various ways:
<ol><li>As limit of infinite tetration: $e=e_3=\max(b\in\R:b[4]\def\e{\infty}\e\in\R)[4]\e$ This b is called the critical base $\def\ {\eta}\ $. It has the value $\ =\ _3=e^{e^{-1}}\approx1.4446678610097661336583391085964$. The function $\ _x$ is not to be confused with the function $\ph_3(x)$ in [https://en.wikipedia.org/wiki/Veblen_function Veblen function], which is also denoted $\ _x$.</li>
<li>As maximum of the self-root $\sqrt[x]x$: e is the argument x, where $\sqrt[x]x$ has its maximum. The maximum is equal to $\ $.</li>
<li>The critical base $\ $ is the base b where the function b^x changes from having 2 fixpoints to having no fixpoint (on the real axis). e is the fixpoint of $\ ^x$.</li></ol>
Now, tetra-Euler and the tetra-critical base is the analogon with using tetration instead of exponentiation (it remains open which extension of tetration we use, and the numeric values may dependently differ, but if not saying differently the candidates are regular tetration for $b\le\ $ and disturbed Fatou coordinates for $b>\ $, which is equal to the Kneser construction).

The tetra-critical base $\ _4\approx1.6353244967152763993453446183062$. 
Tetra-Euler $e_4\approx3.0885322718067176544821807826411$. 
(Values computed by sheldonison.)

It seems we have the same 3 ways of characterization:
<ol><li>Limit of infinite tetration: $\ _4=\max(b\in\R:b[5]\e\in\R)\:e_4=\lim_{n\to\e}b[5]n$ Andrew/Nuninho pointed it out [http://math.eretrandre.org/tetrationforum/showthread.php?tid=269 in this thread].</li>
<li>Maximum of self-tetra-root: The tetraroot is defined as an inverse of tetration. I.e. $y[/4]x=b\iff b[4]x=y$. The self-tetra-root is then defined as x [/4] x. It has its maximum at $x=e_4$ $\ _4=e_4[/4]e_4$. Mike found out in [http://math.eretrandre.org/tetrationforum/showthread.php?tid=468 this thread].</li>
<li>The $\ _4$ is the base where b[4]x changes from having 3 fixpoints to having one fixpoint. $e_4$ is the fixpoint into which the 2 fixpoints transform before they vanish. [http://math.eretrandre.org/tetrationforum/showthread.php?tid=262&pid=5327&mode=threaded Sheldon's posting] contains the following graph and precise values.</li></ol>
[[File:Sexp_fixed.gif]]

A corresponding question/problem about the limit of the Eulers and Etas was asked [http://math.eretrandre.org/tetrationforum/showthread.php?tid=162&pid=5328#pid5328 here].

Euler number

2022-12-12T02:43:32Z

Catullus:

In our context, one can characterize the Euler number e in various ways:
<ol><li>As limit of infinite tetration: $e=e_3=\max(b\in\R:b[4]\infty\in\R)[4]\infty$ This b is called the critical base $\def\ {\eta}\ $. It has the value $\ =\ _3=e^{e^{-1}}\approx1.4446678610097661336583391085964$. The function $\ _x$ is not to be confused with the function $\ph_3(x)$ in [https://en.wikipedia.org/wiki/Veblen_function Veblen function], which is also denoted $\ _x$.</li>
<li>As maximum of the self-root $\sqrt[x]x$: e is the argument x, where $\sqrt[x]x$ has its maximum. The maximum is equal to $\ $.</li>
<li>The critical base $\ $ is the base b where the function b^x changes from having 2 fixpoints to having no fixpoint (on the real axis). e is the fixpoint of $\ ^x$.</li></ol>
Now, tetra-Euler and the tetra-critical base is the analogon with using tetration instead of exponentiation (it remains open which extension of tetration we use, and the numeric values may dependently differ, but if not saying differently the candidates are regular tetration for $b\le\ $ and disturbed Fatou coordinates for $b>\ $, which is equal to the Kneser construction).

The tetra-critical base $\ _4\approx1.6353244967152763993453446183062$. 
Tetra-Euler $e_4\approx3.0885322718067176544821807826411$. 
(Values computed by sheldonison.)

It seems we have the same 3 ways of characterization:
<ol><li>Limit of infinite tetration: $\ _4=\max(b\in\R:b[5]\in\R)\:e_4=\lim_{n\to\infty}b[5]n$ Andrew/Nuninho pointed it out [http://math.eretrandre.org/tetrationforum/showthread.php?tid=269 in this thread].</li>
<li>Maximum of self-tetra-root: The tetraroot is defined as an inverse of tetration. I.e. $y[/4]x=b\iff b[4]x=y$. The self-tetra-root is then defined as x [/4] x. It has its maximum at $x=e_4$ $\ _4=e_4[/4]e_4$. Mike found out in [http://math.eretrandre.org/tetrationforum/showthread.php?tid=468 this thread].</li>
<li>The $\ _4$ is the base where b[4]x changes from having 3 fixpoints to having one fixpoint. $e_4$ is the fixpoint into which the 2 fixpoints transform before they vanish. [http://math.eretrandre.org/tetrationforum/showthread.php?tid=262&pid=5327&mode=threaded Sheldon's posting] contains the following graph and precise values.</li></ol>
[[File:Sexp_fixed.gif]]

A corresponding question/problem about the limit of the Eulers and Etas was asked [http://math.eretrandre.org/tetrationforum/showthread.php?tid=162&pid=5328#pid5328 here].

Euler number

2022-12-12T02:28:28Z

Catullus:

In our context, one can characterize the Euler number e in various ways:
<ol><li>As limit of infinite tetration: $e=e_3=\max(b\in\R:b[4]\infty\in\R)[4]\infty$ This b is called the critical base $\def\ {\eta}\ $. It has the value $\ =\ _3=e^{e^{-1}}\approx1.4446678610097661336583391085964$. The function $\ _x$ is not to be confused with the function $\varphi_3(x)$ in [https://en.wikipedia.org/wiki/Veblen_function Veblen function], which is also denoted $\ _x$.</li>
<li>As maximum of the self-root $\sqrt[x]x$: e is the argument x, where $\sqrt[x]x$ has its maximum. The maximum is equal to $\ $.</li>
<li>The critical base $\ $ is the base b where the function b^x changes from having 2 fixpoints to having no fixpoint (on the real axis). e is the fixpoint of $\ ^x$.</li></ol>
Now, tetra-Euler and the tetra-critical base is the analogon with using tetration instead of exponentiation (it remains open which extension of tetration we use, and the numeric values may dependently differ, but if not saying differently the candidates are regular tetration for $b\le\ $ and disturbed Fatou coordinates for $b>\ $, which is equal to the Kneser construction).

The tetra-critical base $\ _4\approx1.6353244967152763993453446183062$. 
Tetra-Euler $e_4\approx3.0885322718067176544821807826411$. 
(Values computed by sheldonison.)

It seems we have the same 3 ways of characterization:
<ol><li>Limit of infinite tetration: $\ _4=\max(b\in\R:b[5]\in\R)\:e_4=\lim_{n\to\infty}b[5]n$ Andrew/Nuninho pointed it out [http://math.eretrandre.org/tetrationforum/showthread.php?tid=269 in this thread].</li>
<li>Maximum of self-tetra-root: The tetraroot is defined as an inverse of tetration. I.e. $y[/4]x=b\iff b[4]x=y$. The self-tetra-root is then defined as x [/4] x. It has its maximum at $x=e_4$ $\ _4=e_4[/4]e_4$. Mike found out in [http://math.eretrandre.org/tetrationforum/showthread.php?tid=468 this thread].</li>
<li>The $\ _4$ is the base where b[4]x changes from having 3 fixpoints to having one fixpoint. $e_4$ is the fixpoint into which the 2 fixpoints transform before they vanish. [http://math.eretrandre.org/tetrationforum/showthread.php?tid=262&pid=5327&mode=threaded Sheldon's posting] contains the following graph and precise values.</li></ol>
[[File:Sexp_fixed.gif]]

A corresponding question/problem about the limit of the Eulers and Etas was asked [http://math.eretrandre.org/tetrationforum/showthread.php?tid=162&pid=5328#pid5328 here].

User:Catullus

2022-12-11T10:40:37Z

Catullus:

Hello, I am [https://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=310 Catullus]!

Test

2022-12-11T05:36:33Z

Catullus:

This page is for any testing that you want to do.

$x\in\R$ huhu and $z\in\C$

User:Catullus

2022-11-03T03:12:08Z

Catullus:

Hello, I am [https://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=310 Catullus]!

User:Catullus

2022-11-03T02:38:42Z

Catullus: Created page with "I am [https://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=310 Catullus]!"

I am [https://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=310 Catullus]!

Test

2022-11-03T02:00:33Z

Catullus:

This page is for any testing you want to do.

$x\in\R$ huhu and $z\in\C$