If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...
3
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0answers
37 views
Order of Recursion?
Define an extended algebraic function f(a) as a function on a that utilizes any combination of recursive extensions and inversions of sequentiation. Example:
a + 1 = sequentiation. a + a = addition ...
1
vote
0answers
35 views
Tetration Binomial Theorem
I was exploring tetration and came across the following identities:
$${^0}(ab) = 1$$
$${^2}(ab) = ({^2}a)^b * ({^2}b)^a$$
$${^3}(ab) = (ab)^({^2}(ab)) = (ab)^{(({^2}a)^b * ({^2}b)^a)}$$
That third ...
8
votes
1answer
211 views
Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?
(somewhat related to my earlier question)
Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$).
The ...
13
votes
1answer
154 views
Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?
I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
24
votes
1answer
372 views
How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$
I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
0
votes
0answers
53 views
Limits of tetrations of infinite height
We know that tetrations of infinite height converge for $x$ such that $e^{-e} \le x \le e^{1/e}$. Which real numbers are limits of some tetration of infinite height? what is the complete set of such ...
1
vote
0answers
45 views
About growth rate of the iterated exponential on the complex plane.
Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$.
Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$.
How to estimate $S(n,2+i)$ efficiently ?
...
1
vote
0answers
39 views
Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?
Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?
I know this function has 2 other fixed points apart from $0$, so I'm not sure.
Also ...
5
votes
2answers
204 views
Uniqueness of Tetration
Let $f(0)=1$ and $f(x+1)=2^{f(x)}$
Also let f be infinitely differentiable. Then does f exist and is it unique?
If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 ...
6
votes
1answer
264 views
Solutions of $f(f(z)) = e^z$
It is my impression that if we find a function f(z) that satisfies
$$f(f(z)) = e^z $$
there is only one point z that satisfies the relation.
This dawned on me when I noticed that the pesky z that ...
2
votes
3answers
125 views
Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?
In the answering of another question in MSE I've dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic ...
14
votes
2answers
379 views
What is $i$ exponentiated to itself $i$ times?
I was just wondering about this. I searched about it on the net and found that it is called tetration and after this comes pentation and then hexation and so on so forth.
I don't really understand ...
1
vote
1answer
187 views
Find $N$, in the decimal expansion of the large number $N=4^{4^{4^4}}$
Find $N$, in the decimal expansion of the large number
$$N=4^{4^{4^4}}$$
Following on from the question I posted yesterday about finding the number of digits
( Find the number of digits, $D$, in ...
2
votes
4answers
333 views
Find the number of digits, $D$, in the decimal expansion of the large number $N=4^{4^{4^{4}}}$
The full question is:
Find the number of digits, $D$, in the decimal expansion of the large
number
$$N=4^{4^{4^{4}}}$$
Try and find the most efficient ways of finding $D$.
I know that ...
5
votes
0answers
102 views
Notation for n-ary exponentiation
We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator?
$$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
4
votes
1answer
113 views
Operators - sums, products, exponents, etc.
$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$.
$(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$.
Is there an operator, such that if ...
4
votes
1answer
163 views
How to evaluate fractional tetrations?
Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
9
votes
9answers
836 views
half iterate of $x^2+c$
I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial.
$$h(h(x))=x^2$$
$$h(x)=x^{\sqrt{2}}$$
The question is, for $c>0,$ and ...
4
votes
1answer
149 views
Definite integral of tetration between $0$ and $1$
In my old writes I found next formula, where is ${_{}^2}x$ is tetration:
$$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$
And now I am ...
2
votes
2answers
152 views
What is the geometric, physical or other meaning of the tetration?
What is the geometric, physical or other meaning of the tetration or more high hyperoperations?
Is it exists in general or it has only math concept?
2
votes
4answers
233 views
a tetration limit for base $a > e^{1/e}$
Let $a$ be a real number with $a > e^{1/e}$ and $a <> e$.
$slog$ means superlog base $e$ and $sexp$ means superexp base $e$.
$sloga$ means superlog with base $a$ and $sexpa$ means superexp ...
1
vote
5answers
146 views
Limit of a recursively defined bivariate function.
Let m and n be positive integers.
Let $f(m,0)=m$
Let $f(m,n)= e \ln(f(m,n-1))$
$$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$
Where $C$ is a constant.
It seems ...
12
votes
2answers
239 views
A puzzle with powers and tetration mod n
A friend recently asked me if I could solve these three problems:
(a) Prove that the sequence $ 1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of ...
21
votes
1answer
273 views
Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
10
votes
2answers
406 views
Complex towers: $i^{i^{i^{…}}}$
If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
1
vote
2answers
120 views
Find the remainder in the following case where there's a infinite power tower of $7$.
What is the remainder when
$$7^{7^{7^{7^{.^{.^{.^{\infty}}}}}}}$$
is divided by 13?
I'm getting $6$. Is it correct?
10
votes
1answer
262 views
Derivative of $x^{x^{\cdot^{\cdot}}}$?
The infinite tetration is defined as
$$f(x)=x^{x^{\cdot^{\cdot}}}$$
This function is defined for $e^{-e} \leq x \leq e^{e-1}$.
(Wikipedia image)
Can one determine the derivative of this function?
...
4
votes
1answer
180 views
Super logarithmic inverse of tetration
What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$?
Is it $slog^{x}_{2}$?
7
votes
0answers
164 views
Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]
Possible Duplicate:
Infinite tetration, convergence radius
Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that
...
2
votes
3answers
155 views
Mathematical function for the powers
I have this formula $$\underbrace{2^{2^{2^{.^{.^{.^{2^2}}}}}}}_n$$i.e. where the total number of 2's is $n$.
Is there any way to write it as a single mathematical function?
13
votes
1answer
546 views
Infinite tetration, convergence radius
I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework.
The problem is stated as follows. Assume we have an infinite ...
5
votes
0answers
290 views
Is ${^5\pi}$ an integer? [duplicate]
Possible Duplicate:
How to show $e^{e^{e^{79}}}$ is not an integer
Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it?
Here ${^5\pi}$ means the result of tetration ...
39
votes
4answers
2k views
Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?
Problem:
Find $x$ in
$$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$
Trick:
$x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so,
$x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and,
...
7
votes
3answers
456 views
How to calculate $f(x)$ in $f(f(x)) = e^x$?
How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
16
votes
2answers
625 views
Is there a natural way to extend repeated exponentiation beyond integers?
This question has been in my mind since high school.
We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
2
votes
0answers
157 views
What is the Equivalent Form of Tetration to the Exponential $n^{1/n}$?
I've been working on a project for a wiki that I'm a member of. It is the Sequence of the Day for September 2.
You can see my progress at ...
30
votes
10answers
1k views
Why are addition and multiplication commutative, but not exponentiation?
We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why.
As background there are plenty of mathematical schemes that ...
2
votes
1answer
203 views
Could real iterates of the Taylor Series expansion of $b^x$ help to find a way to define tetration?
When we consider the Taylor Series expansion of $f(x)=b^x$ for some $b \in \mathbb{R}$, we see that $$b^x = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}x^n.$$ We can substitute $x$ for $b^x$ to find ...
