Hyperoperations Wiki - User contributions [en]

Tetration-compilation test1

2022-10-30T16:28:48Z

Gottfried:

This page has been copied from a user website in the english wikipedia who has been blocked since 2010 and thus cannot more expanded there. See https://en.wikipedia.org/w/index.php?title=User:MathFacts/Tetration_Summary
---

This is a summary of properties of [[tetration]]. <math>a</math> tetrated to the <math>x</math> is represented by <math>\operatorname{sexp}_a(x)</math> or <math>f_a(x)</math>. The functions <math> \operatorname{log}_a </math> and <math> \operatorname{exp}_a </math> denote [[logarithm]] and [[exponentiation]] to base <math> a </math>, respectively. <math> f^{[k]}(1) </math> is the result of applying the function <math>f</math> to 1 <math> k </math> times in succession. So <math> \exp_a^{[k]}(1) </math>, for example, is equal to <math>\operatorname{sexp}_a(k)</math>, for integer <math>k</math>.

The symbol <math> C_m^k </math> denotes the [[binomial coefficient]], equal to <math>\frac{\Gamma(k+1)}{\Gamma(m+1)\Gamma(k-m+1)}</math>. <math>B_n(x)</math> is the degree <math> n </math> [[Bernoulli polynomial]] of x, while <math>B_n</math> without an argument is the <math>n</math>th [[Bernoulli number]], equal to <math>B_n(0)</math>.

==Evaluation methods==

1. '''Newton method'''. It can be derived from the [[Newton polynomial]] interpolation formula or from the partial iteration theorem.

<math>
\operatorname{sexp}_a(x)=\sum_{m=0}^{\infty} \binom xm \sum_{k=0}^m\,\binom mk\,(-1)^{m-k}\,\operatorname{exp}_a^{[k]}(1)
</math>

This method works for any positive base <math>a\le e^{1/e} \,</math>.

2. '''Lagrange method'''. It can be derived from the [[Lagrange polynomial]] interpolation formula.

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^{n-k}\exp_a^{[k]}(1)
</math>

This method works for positive real bases <math>a\le e^{1/e} \,</math>.

3. '''Rational interpolation method'''

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty}\frac{\sum_{k=0}^{2n} \frac{(-1)^k (k+2)\exp_a^{[k]}(1)}{(x-k) \Gamma (k+1) \Gamma (2n-k+1)}}{\sum_{k=0}^{2n} \frac{(-1)^k (k+2)}{(x-k) \Gamma (k+1) \Gamma (2n-k+1)}}
</math>

This method works for positive real bases <math>a\le e^{1/e} \,</math>.

4. '''Regular iteration method'''. It is derived using the technique of [[regular iteration]] at the fixed point of the exponential function.

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty} \log_a^{[n]}\left(\left(1-\left(\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\right)^x\right)\frac{W(-\ln a)}{-\ln a}+\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\exp_a^{[n]}(1)\right)
</math>

<math>
\operatorname{sexp}_a(x)=\left(\lim_{n\to\infty} \frac{\ln \left(\frac{\frac{W(-\ln a )}{\ln a}+\exp_a^{[n]}(x)}{\frac{W(-\ln a)}{\ln a}+\exp_a^{[n]}(1)}\right)}{\ln \ln \left(\frac{W(-\ln a)}{-\ln a}\right)}\right)^{[-1]}
</math>

Here the W function is the product logarithm or [[Lambert W function]], which is defined by <math>W(x) \exp(W(x)) = x\,</math>. The expression <math> \frac{W(-\ln a)}{-\ln a} </math> is the principal fixed point of the function <math>a^x</math>.

This method works for real bases <math>e^{-1/e} < a\le e^{1/e} \,</math>.

5. '''Matrix power method'''. One can use [[Carleman matrix|Carleman matrices]] to find the iterates<ref>http://math.eretrandre.org/tetrationforum/attachment.php?aid=318</ref>.

This method works for real bases <math>a>1 \,</math>.

6. '''Cauchy integral iteration method''': A method developed by Dmitriy Kouznetsov <ref name="k">Dmitrii Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>.

One should apply the [[Cauchy integral formula]] to the loop around the rectangle from <math> n-1 - K i </math> to <math> n+1 + K i </math>, where <math>n</math> is the integer part of <math>x</math>, and <math>K </math> is substantially larger than the real or imaginary parts of <math>x</math>. The value of the integrands along the imaginary-axis edges of this rectangle can be determined from their value along the imaginary line through <math>n</math>, while the value along the real-axis edges of the rectangle can be estimated under the assumption that the function tends to the two (complex) fixed points of <math> x = a^x </math> at infinite imaginary values.

To obtain the value of the function along the imaginary line through <math>n</math>, we start with a guess value for this function, and recursively correct it using the Cauchy formula.

This method works for real bases <math> a \ge e^{1/e} \,</math>.

7. '''Intuitive iteration method''' (formerly called "natural" method, or matrix inverse method). A method first developed by Peter Walker, then rediscovered by Andrew Robbins, which uses matrices to solve the Abel functional equation for exponentials:

:<math>\alpha(b^x) = \alpha(x) + 1\,</math>

applying the Carleman matrix to both sides, and simplifying the matrices a bit, we are left with the matrix equation:

:<math>(C[b^x]^T - I)D[\alpha] = D[1]\,</math>

where C is a Carleman matrix, and D is a Taylor coefficient column vector. Since this equation is linear, we can solve for the Taylor coefficients of the Abel function <math>\alpha(x)</math> making <math>(C[b^x]^T - I)</math> invertible (by truncating the first column and last row). This truncation is called the Abel matrix:

:<math>A[b^x] = J(C[b^x]^T - I)K\,</math>

where J is the identity matrix without the last row, and K is the identity matrix without the first column. The importance of the Abel matrix is that it is often invertible, in which case we can solve for the Taylor coefficients of <math>\alpha(x)\,</math> as <math>D[\alpha] = A[b^x]^{-1}D[1]\,</math>.

To summarize, the super-logarithm is the Abel function of exponentials, so the Taylor coefficients of the super-logarithm can be found in the first column of the inverse of the Abel matrix.

This method works for real bases <math>a>1 \,</math>.

==Functional and differential equations==

1. '''Main functional equation'''

<math>f_a(x+1)=a^{f_a(x)}</math>

2. '''Main functional equation for inverse function'''

<math>f^{[-1]}_a (b) = 1 + f^{[-1]}_a (\log_a b) \,</math>

3. '''Differential-difference equation'''

<math>f'_a(x+1)=f'_a(x)f_a(x+1)\ln a \,</math>

==Periodicity==

Superexponential with fixed base ''b'' is periodic<ref>[http://www.ils.uec.ac.jp/~dima/PAPERS/2009sqrt2.pdf Dmitrii Kouznetsov. Portrait of the four regular super- exponentials to base sqrt(2)]</ref> with period
:<math>T=\frac{2\pi i }{\ln \ln \frac{W(-\ln b)}{-\ln b}}</math>

==Values in fixed points and asymptotic properties==

* <math>f_a(0)=1 \,</math>
* <math>f_a(-1)=0 \,</math>
* <math>f_a(1)=a \,</math>
* <math>f_a(-2)=-\frac{\infty}{\ln(a)}</math>
* <math>f_a(+\infty)=-\frac{\mathrm{W}(-\ln{a})}{\ln{a}}</math>
* <math>f'_e(-1)=f'_e(0) \,</math>

==Other properties==

<math>
\log_a \frac{f'_a(x)}{f'_a(0)(\ln a)^{x}} = \sum_{k=0}^{x-1} f_a(k)
</math>

<math>
\frac{f'_a(x)}{f'_a(0)(\ln a)^x}= \prod_{k=0}^x f_a(k)
</math>

<math>\log_a \frac{f'_a(x)}{f'_a(0)(\ln a)^{x}} = \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} (B_n(x)-B_n)</math>

==Differentiation rules==

1. '''Differentiating tetration with fixed height'''

:<math>(\operatorname{sexp}_x(1))'=1</math>
:<math>(\operatorname{sexp}_x(n))'=\left((\operatorname{sexp}_x(n-1))'\ln x + \frac{\operatorname{sexp}_x(n-1)}{x}\right)\operatorname{sexp}_x(n)</math>

Generalized,
:<math>(\text{sexp}_x(n))'=\frac1x \sum _{k=1}^n (\ln x)^{k-1}\prod _{j=n-k}^n \text{sexp}_x(j)</math>

2. '''Differentiating tetration with fixed base'''

:<math>\operatorname{sexp}'_a(x)=\operatorname{sexp}'_a(x-1)\operatorname{sexp}_a(x)\ln a \,</math>

Generalized,
:<math>(\operatorname{sexp}_a(x))' =\operatorname{sexp}'_a(0)(\ln a)^x \prod_{j=1}^x \operatorname{sexp}_a(j)</math>

:<math>(\operatorname{sexp}_a(x))' =\operatorname{sexp}'_a(c)(\ln a)^{x - c} \!\prod_{j=c+1}^x\! \operatorname{sexp}_a(j)</math>

==Approximation methods==

In light of the main equation above, it suffices to define an approximation function on an interval of unit length.

# '''Linear approximation''': On the interval [-1, 0], we have <math> \operatorname{sexp}_a(x) \approx x+1 </math>. This approximation is continuous everywhere, but generally not differentiable at integers.
# '''Quadratic approximation''': On the interval [-1, 0], we have <math> \operatorname{sexp}_a(x) \approx {\log(a) - 1 \over 1 + \log(a)} x^2+ {2 \log(a) \over 1+\log(a)} x+1 </math>. This approximation is continuously differentiable everywhere, but its second derivative is discontinuous at each integer.
# '''Shifted linear approximation''': Developed by Jay D. Fox at [http://math.eretrandre.org/tetrationforum/showthread.php?tid=98]. On some interval [c, c+1], we have <math> \operatorname{sexp}_a(x) \approx {1 + \log(\log(a)) \over \log(a)} (x-c) - {\log(\log(a)) \over \log(a)} </math>. The value c is chosen so that the approximation gives <math> \operatorname{sexp}_a(0) \approx 1</math>; if <math> a \ge e </math>, this means setting <math> c = -1 - {\log(\log(a)) \over 1 + \log(\log(a))} </math>, while if <math> a \le e \le a^a </math>, it means setting <math> c = {-\log(a) - \log(\log(a)) \over 1 + \log(\log(a))} </math>. This approximation is continuously differentiable everywhere, but its second derivative is discontinuous at ''c''+''n'' for any integer ''n''. It is only defined for <math> a > e^{1/e} </math>.

If <math> a = e </math> all three of the above approximations are equivalent.

===Approximate values===

* For base 10: 
:{| class="wikitable"
! Approximation
! <math> \operatorname{sexp}_{10}(1/2) </math>
! <math> \operatorname{slog}_{10}(2) </math>
|-
|Linear: <math>\,{}^{x}10 \approx 10^x</math> for <math>0<x<1</math>
| 3.162776601...
| 0.301029995...
|-
|Quadratic: <math>\,{}^{x}10 \approx {}^{x}a \approx 10^{x + \frac{-1+\log(10)}{1+\log(10)}(x^2-x)} = 10^{x + 0.39441379... (x^2 - x)} </math> for <math>0<x<1</math>
| 2.5199768...
|
|-
|Shifted Linear: <math>\,{}^{x}10 \approx .79651... (x+1) </math> for <math>-1.454753...<x<-0.454753...</math>
| 2.50181...
| 0.377936...
|}

==References==

<references/>

Tetration-compilation test1

2022-10-30T16:26:07Z

Gottfried: Created page with "This page has been copied from a blocked user website in the english wikipedia https://en.wikipedia.org/w/index.php?title=User:MathFacts/Tetration_Summary&action=edit This i..."

This page has been copied from a blocked user website in the english wikipedia https://en.wikipedia.org/w/index.php?title=User:MathFacts/Tetration_Summary&action=edit

This is a summary of properties of [[tetration]]. <math>a</math> tetrated to the <math>x</math> is represented by <math>\operatorname{sexp}_a(x)</math> or <math>f_a(x)</math>. The functions <math> \operatorname{log}_a </math> and <math> \operatorname{exp}_a </math> denote [[logarithm]] and [[exponentiation]] to base <math> a </math>, respectively. <math> f^{[k]}(1) </math> is the result of applying the function <math>f</math> to 1 <math> k </math> times in succession. So <math> \exp_a^{[k]}(1) </math>, for example, is equal to <math>\operatorname{sexp}_a(k)</math>, for integer <math>k</math>.

The symbol <math> C_m^k </math> denotes the [[binomial coefficient]], equal to <math>\frac{\Gamma(k+1)}{\Gamma(m+1)\Gamma(k-m+1)}</math>. <math>B_n(x)</math> is the degree <math> n </math> [[Bernoulli polynomial]] of x, while <math>B_n</math> without an argument is the <math>n</math>th [[Bernoulli number]], equal to <math>B_n(0)</math>.

==Evaluation methods==

1. '''Newton method'''. It can be derived from the [[Newton polynomial]] interpolation formula or from the partial iteration theorem.

<math>
\operatorname{sexp}_a(x)=\sum_{m=0}^{\infty} \binom xm \sum_{k=0}^m\,\binom mk\,(-1)^{m-k}\,\operatorname{exp}_a^{[k]}(1)
</math>

This method works for any positive base <math>a\le e^{1/e} \,</math>.

2. '''Lagrange method'''. It can be derived from the [[Lagrange polynomial]] interpolation formula.

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^{n-k}\exp_a^{[k]}(1)
</math>

This method works for positive real bases <math>a\le e^{1/e} \,</math>.

3. '''Rational interpolation method'''

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty}\frac{\sum_{k=0}^{2n} \frac{(-1)^k (k+2)\exp_a^{[k]}(1)}{(x-k) \Gamma (k+1) \Gamma (2n-k+1)}}{\sum_{k=0}^{2n} \frac{(-1)^k (k+2)}{(x-k) \Gamma (k+1) \Gamma (2n-k+1)}}
</math>

This method works for positive real bases <math>a\le e^{1/e} \,</math>.

4. '''Regular iteration method'''. It is derived using the technique of [[regular iteration]] at the fixed point of the exponential function.

<math>
\operatorname{sexp}_a(x)=\lim_{n\to\infty} \log_a^{[n]}\left(\left(1-\left(\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\right)^x\right)\frac{W(-\ln a)}{-\ln a}+\ln \left(\frac{W(-\ln a)}{-\ln a}\right)\exp_a^{[n]}(1)\right)
</math>

<math>
\operatorname{sexp}_a(x)=\left(\lim_{n\to\infty} \frac{\ln \left(\frac{\frac{W(-\ln a )}{\ln a}+\exp_a^{[n]}(x)}{\frac{W(-\ln a)}{\ln a}+\exp_a^{[n]}(1)}\right)}{\ln \ln \left(\frac{W(-\ln a)}{-\ln a}\right)}\right)^{[-1]}
</math>

Here the W function is the product logarithm or [[Lambert W function]], which is defined by <math>W(x) \exp(W(x)) = x\,</math>. The expression <math> \frac{W(-\ln a)}{-\ln a} </math> is the principal fixed point of the function <math>a^x</math>.

This method works for real bases <math>e^{-1/e} < a\le e^{1/e} \,</math>.

5. '''Matrix power method'''. One can use [[Carleman matrix|Carleman matrices]] to find the iterates<ref>http://math.eretrandre.org/tetrationforum/attachment.php?aid=318</ref>.

This method works for real bases <math>a>1 \,</math>.

6. '''Cauchy integral iteration method''': A method developed by Dmitriy Kouznetsov <ref name="k">Dmitrii Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>.

One should apply the [[Cauchy integral formula]] to the loop around the rectangle from <math> n-1 - K i </math> to <math> n+1 + K i </math>, where <math>n</math> is the integer part of <math>x</math>, and <math>K </math> is substantially larger than the real or imaginary parts of <math>x</math>. The value of the integrands along the imaginary-axis edges of this rectangle can be determined from their value along the imaginary line through <math>n</math>, while the value along the real-axis edges of the rectangle can be estimated under the assumption that the function tends to the two (complex) fixed points of <math> x = a^x </math> at infinite imaginary values.

To obtain the value of the function along the imaginary line through <math>n</math>, we start with a guess value for this function, and recursively correct it using the Cauchy formula.

This method works for real bases <math> a \ge e^{1/e} \,</math>.

7. '''Intuitive iteration method''' (formerly called "natural" method, or matrix inverse method). A method first developed by Peter Walker, then rediscovered by Andrew Robbins, which uses matrices to solve the Abel functional equation for exponentials:

:<math>\alpha(b^x) = \alpha(x) + 1\,</math>

applying the Carleman matrix to both sides, and simplifying the matrices a bit, we are left with the matrix equation:

:<math>(C[b^x]^T - I)D[\alpha] = D[1]\,</math>

where C is a Carleman matrix, and D is a Taylor coefficient column vector. Since this equation is linear, we can solve for the Taylor coefficients of the Abel function <math>\alpha(x)</math> making <math>(C[b^x]^T - I)</math> invertible (by truncating the first column and last row). This truncation is called the Abel matrix:

:<math>A[b^x] = J(C[b^x]^T - I)K\,</math>

where J is the identity matrix without the last row, and K is the identity matrix without the first column. The importance of the Abel matrix is that it is often invertible, in which case we can solve for the Taylor coefficients of <math>\alpha(x)\,</math> as <math>D[\alpha] = A[b^x]^{-1}D[1]\,</math>.

To summarize, the super-logarithm is the Abel function of exponentials, so the Taylor coefficients of the super-logarithm can be found in the first column of the inverse of the Abel matrix.

This method works for real bases <math>a>1 \,</math>.

==Functional and differential equations==

1. '''Main functional equation'''

<math>f_a(x+1)=a^{f_a(x)}</math>

2. '''Main functional equation for inverse function'''

<math>f^{[-1]}_a (b) = 1 + f^{[-1]}_a (\log_a b) \,</math>

3. '''Differential-difference equation'''

<math>f'_a(x+1)=f'_a(x)f_a(x+1)\ln a \,</math>

==Periodicity==

Superexponential with fixed base ''b'' is periodic<ref>[http://www.ils.uec.ac.jp/~dima/PAPERS/2009sqrt2.pdf Dmitrii Kouznetsov. Portrait of the four regular super- exponentials to base sqrt(2)]</ref> with period
:<math>T=\frac{2\pi i }{\ln \ln \frac{W(-\ln b)}{-\ln b}}</math>

==Values in fixed points and asymptotic properties==

* <math>f_a(0)=1 \,</math>
* <math>f_a(-1)=0 \,</math>
* <math>f_a(1)=a \,</math>
* <math>f_a(-2)=-\frac{\infty}{\ln(a)}</math>
* <math>f_a(+\infty)=-\frac{\mathrm{W}(-\ln{a})}{\ln{a}}</math>
* <math>f'_e(-1)=f'_e(0) \,</math>

==Other properties==

<math>
\log_a \frac{f'_a(x)}{f'_a(0)(\ln a)^{x}} = \sum_{k=0}^{x-1} f_a(k)
</math>

<math>
\frac{f'_a(x)}{f'_a(0)(\ln a)^x}= \prod_{k=0}^x f_a(k)
</math>

<math>\log_a \frac{f'_a(x)}{f'_a(0)(\ln a)^{x}} = \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} (B_n(x)-B_n)</math>

==Differentiation rules==

1. '''Differentiating tetration with fixed height'''

:<math>(\operatorname{sexp}_x(1))'=1</math>
:<math>(\operatorname{sexp}_x(n))'=\left((\operatorname{sexp}_x(n-1))'\ln x + \frac{\operatorname{sexp}_x(n-1)}{x}\right)\operatorname{sexp}_x(n)</math>

Generalized,
:<math>(\text{sexp}_x(n))'=\frac1x \sum _{k=1}^n (\ln x)^{k-1}\prod _{j=n-k}^n \text{sexp}_x(j)</math>

2. '''Differentiating tetration with fixed base'''

:<math>\operatorname{sexp}'_a(x)=\operatorname{sexp}'_a(x-1)\operatorname{sexp}_a(x)\ln a \,</math>

Generalized,
:<math>(\operatorname{sexp}_a(x))' =\operatorname{sexp}'_a(0)(\ln a)^x \prod_{j=1}^x \operatorname{sexp}_a(j)</math>

:<math>(\operatorname{sexp}_a(x))' =\operatorname{sexp}'_a(c)(\ln a)^{x - c} \!\prod_{j=c+1}^x\! \operatorname{sexp}_a(j)</math>

==Approximation methods==

In light of the main equation above, it suffices to define an approximation function on an interval of unit length.

# '''Linear approximation''': On the interval [-1, 0], we have <math> \operatorname{sexp}_a(x) \approx x+1 </math>. This approximation is continuous everywhere, but generally not differentiable at integers.
# '''Quadratic approximation''': On the interval [-1, 0], we have <math> \operatorname{sexp}_a(x) \approx {\log(a) - 1 \over 1 + \log(a)} x^2+ {2 \log(a) \over 1+\log(a)} x+1 </math>. This approximation is continuously differentiable everywhere, but its second derivative is discontinuous at each integer.
# '''Shifted linear approximation''': Developed by Jay D. Fox at [http://math.eretrandre.org/tetrationforum/showthread.php?tid=98]. On some interval [c, c+1], we have <math> \operatorname{sexp}_a(x) \approx {1 + \log(\log(a)) \over \log(a)} (x-c) - {\log(\log(a)) \over \log(a)} </math>. The value c is chosen so that the approximation gives <math> \operatorname{sexp}_a(0) \approx 1</math>; if <math> a \ge e </math>, this means setting <math> c = -1 - {\log(\log(a)) \over 1 + \log(\log(a))} </math>, while if <math> a \le e \le a^a </math>, it means setting <math> c = {-\log(a) - \log(\log(a)) \over 1 + \log(\log(a))} </math>. This approximation is continuously differentiable everywhere, but its second derivative is discontinuous at ''c''+''n'' for any integer ''n''. It is only defined for <math> a > e^{1/e} </math>.

If <math> a = e </math> all three of the above approximations are equivalent.

===Approximate values===

* For base 10: 
:{| class="wikitable"
! Approximation
! <math> \operatorname{sexp}_{10}(1/2) </math>
! <math> \operatorname{slog}_{10}(2) </math>
|-
|Linear: <math>\,{}^{x}10 \approx 10^x</math> for <math>0<x<1</math>
| 3.162776601...
| 0.301029995...
|-
|Quadratic: <math>\,{}^{x}10 \approx {}^{x}a \approx 10^{x + \frac{-1+\log(10)}{1+\log(10)}(x^2-x)} = 10^{x + 0.39441379... (x^2 - x)} </math> for <math>0<x<1</math>
| 2.5199768...
|
|-
|Shifted Linear: <math>\,{}^{x}10 \approx .79651... (x+1) </math> for <math>-1.454753...<x<-0.454753...</math>
| 2.50181...
| 0.377936...
|}

==References==

<references/>

Help:LaTeX

2017-10-08T00:58:12Z

Gottfried: /* Display style formulas */

== Inline formulas ==
<pre>$..\sqrt 3 .$</pre>

== Display style formulas ==
<pre>$$..\sqrt 3.$$</pre>

== Latex macros ==
You can use the following abbreviations:
The following list is an excerpt from an initialization file of MathJax and is to be understand, e.g. the first line, as: Use <pre>\C</pre> for $\C$.
<pre>
C: '\\mathbb{C}', /* the complex numbers */
N: '\\mathbb{N}', /* the natural numbers */
Q: '\\mathbb{Q}', /* the rational numbers */
R: '\\mathbb{R}', /* the real numbers */
Z: '\\mathbb{Z}', /* the integer numbers */
ph: '\\varphi',
eps: '\\varepsilon',
th: '\\vartheta',

/* some extre macros for ease of use; these are non-standard! */
F: '\\mathbb{F}', /* a finite field */
HH: '\\mathcal{H}', /* a Hilbert space */
bszero: '\\boldsymbol{0}', /* vector of zeros */
bsone: '\\boldsymbol{1}', /* vector of ones */
bst: '\\boldsymbol{t}', /* a vector 't' */
bsv: '\\boldsymbol{v}', /* a vector 'v' */
bsw: '\\boldsymbol{w}', /* a vector 'w' */
bsx: '\\boldsymbol{x}', /* a vector 'x' */
bsy: '\\boldsymbol{y}', /* a vector 'y' */
bsz: '\\boldsymbol{z}', /* a vector 'z' */
bsDelta: '\\boldsymbol{\\Delta}', /* a vector '\Delta' */
E: '\\mathrm{e}', /* the exponential */
rd: '\\,\\mathrm{d}', /* roman d for use in integrals: $\int f(x) \rd x$ */
rdelta: '\\,\\delta', /* delta operator for use in sums */
rD: '\\mathrm{D}', /* differential operator D */

/* example from MathJax on how to define macros with parameters: */
/* bold: ['{\\bf #1}', 1] */

RR: '\\mathbb{R}',
ZZ: '\\mathbb{Z}',
NN: '\\mathbb{N}',
QQ: '\\mathbb{Q}',
CC: '\\mathbb{C}',
FF: '\\mathbb{F}'
</pre>

Help:LaTeX

2017-10-08T00:57:25Z

Gottfried: /* Inline formulas */

== Inline formulas ==
<pre>$..\sqrt 3 .$</pre>

== Display style formulas ==
<pre>$$...$$</pre>

== Latex macros ==
You can use the following abbreviations:
The following list is an excerpt from an initialization file of MathJax and is to be understand, e.g. the first line, as: Use <pre>\C</pre> for $\C$.
<pre>
C: '\\mathbb{C}', /* the complex numbers */
N: '\\mathbb{N}', /* the natural numbers */
Q: '\\mathbb{Q}', /* the rational numbers */
R: '\\mathbb{R}', /* the real numbers */
Z: '\\mathbb{Z}', /* the integer numbers */
ph: '\\varphi',
eps: '\\varepsilon',
th: '\\vartheta',

/* some extre macros for ease of use; these are non-standard! */
F: '\\mathbb{F}', /* a finite field */
HH: '\\mathcal{H}', /* a Hilbert space */
bszero: '\\boldsymbol{0}', /* vector of zeros */
bsone: '\\boldsymbol{1}', /* vector of ones */
bst: '\\boldsymbol{t}', /* a vector 't' */
bsv: '\\boldsymbol{v}', /* a vector 'v' */
bsw: '\\boldsymbol{w}', /* a vector 'w' */
bsx: '\\boldsymbol{x}', /* a vector 'x' */
bsy: '\\boldsymbol{y}', /* a vector 'y' */
bsz: '\\boldsymbol{z}', /* a vector 'z' */
bsDelta: '\\boldsymbol{\\Delta}', /* a vector '\Delta' */
E: '\\mathrm{e}', /* the exponential */
rd: '\\,\\mathrm{d}', /* roman d for use in integrals: $\int f(x) \rd x$ */
rdelta: '\\,\\delta', /* delta operator for use in sums */
rD: '\\mathrm{D}', /* differential operator D */

/* example from MathJax on how to define macros with parameters: */
/* bold: ['{\\bf #1}', 1] */

RR: '\\mathbb{R}',
ZZ: '\\mathbb{Z}',
NN: '\\mathbb{N}',
QQ: '\\mathbb{Q}',
CC: '\\mathbb{C}',
FF: '\\mathbb{F}'
</pre>

Shell-Thron region

2011-06-21T10:57:07Z

Gottfried:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
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
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

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:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
\end{align*}$$

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>
The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].

[[File:Shell-region.png]]

<hr>

This picture 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:ShellThron_utb.png|600px]]

== 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=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shell-Thron region

2011-06-21T10:50:12Z

Gottfried: /* Relevant posts on the Tetrationforum */

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
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
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

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:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
[[File:Shell-region.png]]

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:ShellThron_utb.png]]

== 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=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shell-Thron region

2011-06-21T10:47:53Z

Gottfried: /* Relevant posts on the Tetrationforum */

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
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
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

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:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
[[File:Shell-region.png]]

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:ShellThron_utb.png]]

== 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=120 Infinite tetration of the imaginary unit]

Shell-Thron region

2011-06-21T10:45:25Z

Gottfried:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
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
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

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:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
[[File:Shell-region.png]]

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:ShellThron_utb.png]]

== Relevant posts on the Tetrationforum ==

[http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

Shell-Thron region

2011-06-21T10:43:13Z

Gottfried:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
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
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

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:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].
[[File:Shell-region.png]]

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:
[[File:ShellThron_utb.png]]

== Relevant posts on the Tetrationforum ==

[http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

File:ShellThron utb.png

2011-06-21T10:34:36Z

Gottfried: Combined view of the Shell-Thron-region of infinite powertower in the complex plane of fixpoint, corresponding base and log of fixpoint

Combined view of the Shell-Thron-region of infinite powertower in the complex plane of fixpoint, corresponding base and log of fixpoint

User:Gottfried

2011-06-03T13:34:34Z

Gottfried: Created page with "Gotti dummy text"

Gotti
dummy text