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# Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

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139 views

### Are there infinitely many $c$-th perfect powers having a constant congruence speed of $c$?

Let $a$ be a positive integer of the form $20 \cdot n + 5$ (i.e., $a : a \equiv 5 \pmod {20}$, $n \in \mathbb{N}_0$). I wish to prove (or disporove) the following statement. Let $c \in \mathbb{Z}^+$ ...
83 views

### How can the domains of all variables of this hyperoperation function be extended to the entire complex plane?

When generalizing addition, multiplication, and exponentiation, there exists a certain hierarchy upon which these operators can be placed: the hyperoperator hierarchy. Starting with succession, each ...
1 vote
72 views

### Iterating $\tanh(A x)$ and $\lim_{x \to +\infty} \tanh^{[r]}(A x) = C_r$?

Everybody who ever studied special relativity or hyperbolic trig knows this function $$\tanh(Ax)$$ for real $0 < A < 1$ $\tanh(A x)$ has a nice attracting fixpoint at $0$ and it is asymptotic to ...
79 views

### Evaluating $\frac{d^n}{dt^n}e^{a(t-e^t)}$ as a single series to extend region of convergence for super root function

$\def\srt{\operatorname{srt}}$Introduction: There is a multiple series expansion for the super root $\srt_n(z)$ valid near $0.7<|z|<1.4$. However, for around $0<|z|<1.3$, there is this ...
53 views

### How to find what number raised to its own power is equal to a given number? [duplicate]

If $x^x = 3$, how can I find x? I know I can rearrange this to $\log_x(3) = x$, and that some calculators can solve this, but how would you do this manually?
62 views

### Closed form for $H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-k, j)j^k$ where $H(1,k)=\frac{1}{k!}$

Let $H(n,k)$ be defined such that $$H(1,k)=\frac{1}{k!}\text{, and }H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-1,j)j^k$$ As pointed out in the comments, I should mention that we must define $0^0=1$ as ...
59 views

1 vote
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### What is $^{y(slog_xa)}x$?

What is $^{y(slog_xa)}x$ ? In this website about tetration, it's shown that $^{y+slog_xa}x=$ $^{y}x^a$, as $^{1+slog_xa}x=x^a$. So what is $^{y(slog_xa)}x$ ? Is it $^{y}a$ ?
1 vote
90 views

### Negative Tetrations?

To start, I'll say that for this post I'll be using Rudy Rucker notation for tetration. That being $^2$x=$x^x%$, which means the number raised to the left means how many times one would exponentiate x....
70 views

### General formula for the derivative of a pentation?

Now, many people probably know of the first three hyperoperations, such as addition, multiplication, and exponentiation. However, many don't know of the fourth, tetration, and even less then know ...
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### Can we characterize functions, like exponentials, the gamma function, and tetration, as solutions of an optimization problem?

This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to ...
172 views

### General Rule for Differentiation of Tetrations

I'll start at the beginning. Initially, this sort of began as just what is $\frac{d}{dx}$[$x^x$] the answer being $x^x$+ln(x)$x^x$. This wasn't difficult to achieve, just some chain rule and product ...
1 vote
118 views

### Does anyone know if it's possible to solve $x^x=x+1$ in terms of $x$?

So I have tried solving for $x$ algebraicly using the productlog function but all I was able to do is: $$x\log(x) = W(x\log(x)(x+1))$$ Maybe I could use the square-super root formula $e^{W(\log(x))}$, ...
1 vote
195 views

### Is field of complex numbers closed under tetration?

The set of real numbers is closed under multiplication, but not under exponentiation (Eg. square root of negative numbers). That is, $\exists a, b \in R \mid {a^b} \notin R$. Then we introduced ...
1 vote
84 views

### Can we generalize the tetration to the real or complex numbers? [duplicate]

is it possible to find a value for this operation: ${^{(3/2)}2}$? If so, can we generalize the domain of the function ${^{x}a}$ to the real or even complex numbers? I had originally tried to solve the ...
Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ...