# 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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### 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....
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### 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 ...
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### 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
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### What is the equivalent of the factorial function, but for exponentiation?

The factorial function multiplies a given number by each number less than itself until reaching one. Does a function, notation, or literature yet exist regarding the idea of raising a given number to ...
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### 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
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### 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
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### 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 ...