Euler number - Revision history

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Bo198214: Created page with "In our context one can characterize the Euler number e in various ways:

As limit of infinite tetration: $e=e_3$ is the limit b[4]oo for the maximal b such that the lim..."

2011-06-02T22:30:26Z

Created page with "In our context one can characterize the Euler number e in various ways: <ol> <li> As limit of infinite tetration: $e=e_3$ is the limit b[4]oo for the maximal b such that the lim..."

New page

In our context one can characterize the Euler number e in various ways:

<ol>
<li> As limit of infinite tetration: $e=e_3$ is the limit b[4]oo for the maximal b such that the limit exists. This b is called the <i>critical base</i> $\eta$, it has the value $\eta=\eta_3=e^{1/e}$.</li>

<li>As maximum of the self-root $x^{1/x}$: e is the argument x, where $x^{1/x}$ has its maximum, the maximum is equal to $\eta$.</li>

<li>The critical base $\eta$ 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 $\eta^x$.</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 $b\le e^{1/e}$ and disturbed Fatou coordinates for $b>e^{1/e}$ which is equal to the Kneser construction).

The tetra-critical base $\eta_4 \approx 1.6353244967152763993453446183062$.<br/>
Tetra-Euler $e_4\approx 3.0885322718067176544821807826411$.<br/>
(Values computed by sheldonison.)

It seems we have the same 3 ways of characterization:

<ol>
<li> Limit of infinite tetration: $\eta_4$ is the greatest number $b$ such that b[5]oo exists, $e_4=\lim_{n\to\infty} b[5]n$.
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 the inverse of tetration, i.e. if b[4]x=y then y [/4] x = b. The self-tetra-root is then defined as x [/4] x. It has its maximum at $x=e_4$ and $\eta_4 = e_4 [/4] e_4$.
Mike found out in [http://math.eretrandre.org/tetrationforum/showthread.php?tid=468 this thread].</li>

<li> The $\eta_4$ 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:http://math.eretrandre.org/tetrationforum/attachment.php?aid=796]]

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].

@@ Line 1: / Line 1: @@
 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 <i>critical base</i> $\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>
+<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 <i>critical base</i> $\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>
@@ Line 10: / Line 10: @@
 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>
+<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>

@@ Line 1: / Line 1: @@
 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 <i>critical base</i> $\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>
+<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 <i>critical base</i> $\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>

@@ Line 1: / Line 1: @@
-In our context one can characterize the Euler number e in various ways:
+In our context, one can characterize the Euler number e in various ways:
-<ol>
+The tetra-critical base $\ _4\approx1.6353244967152763993453446183062$.<br/>
-<li> As limit of infinite tetration: $e=e_3$ is the limit b[4]oo for the maximal b such that the limit exists. This b is called the <i>critical base</i> $\eta$, it has the value $\eta=\eta_3=e^{1/e}$.</li>
+Tetra-Euler $e_4\approx3.0885322718067176544821807826411$.<br/>
 (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>
-<ol>
+<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> Limit of infinite tetration: $\eta_4$ is the greatest number $b$ such that b[5]oo exists, $e_4=\lim_{n\to\infty} b[5]n$.
+<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].

@@ Line 27: / Line 27: @@
 </ol>
-[[File:http://math.eretrandre.org/tetrationforum/attachment.php?aid=796]]
+http://math.eretrandre.org/tetrationforum/attachment.php?aid=796
 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].