Shell-Thron region - Revision history

Bo198214 at 19:09, 16 July 2019

2019-07-16T19:09:38Z

2019-07-16T19:08:44Z

2019-07-16T19:08:08Z

2019-07-16T18:41:59Z

2011-06-21T10:57:07Z

2011-06-21T10:50:12Z

Relevant posts on the Tetrationforum

2011-06-21T10:47:53Z

Relevant posts on the Tetrationforum

2011-06-21T10:45:25Z

2011-06-21T10:43:13Z

2011-06-04T17:54:02Z

explicating

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 These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.
-Following from this the boundary can be given as a curve $\phi\mapsto \exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.
+Following from this the boundary can be given as a curve $\phi\mapsto \exp(e^{i\phi-e^{i\phi}}),\phi\in[0,2\pi)$.
 <hr>

@@ Line 15: / Line 15: @@
 These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.
-Following from this the boundary can be given as a curve $\phi\mapsto exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.
+Following from this the boundary can be given as a curve $\phi\mapsto \exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.
 <hr>

@@ Line 15: / Line 15: @@
 These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.
-The boundary can also be given by the formula $|-W(-\log(b))|=1$.
+Following from this the boundary can be given as a curve $\phi\mapsto exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.
 <hr>

@@ Line 3: / Line 3: @@
 exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.
-If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$ and $b=\lambda^{1/\lambda}$, or
+If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$, or
 $$\lambda=\frac{W(-\log(b))}{-\log(b)}$$
 By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.
-The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ due to the following rearrangements:
+The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ (i.e. lambda is a parabolic fixpoint of $\exp_b$) due to the following rearrangements:
 $$\begin{align*}
-=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
+=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(b^\lambda)|=|\log(\lambda)|
 \end{align*}$$
 These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.
 <hr>

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 These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.
-The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
+<hr>
-[[File:Shell-region.png]]
+The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
-Here is another picture which gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )
+[[File:Shell-region.png]]
 [[File:ShellThron_utb.png]]
 == Relevant posts on the Tetrationforum ==

@@ Line 23: / Line 23: @@
 == Relevant posts on the Tetrationforum ==
-[http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]
+*  [http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]
 [http://math.eretrandre.org/tetrationforum/showthread.php?tid=120 Infinite tetration of the imaginary unit]

← Older revision		Revision as of 10:47, 21 June 2011
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	[http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]		[http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]
		+	[http://math.eretrandre.org/tetrationforum/showthread.php?tid=120 Infinite tetration of the imaginary unit]