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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/>

@@ Line 1: / Line 1: @@
-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 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>.

Tetration-compilation test1 - Revision history

Gottfried at 16:28, 30 October 2022

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..."