Questions tagged [tetration]
By Daniel Rodriguez •
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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.... calculus derivatives tetration- 93
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 ... calculus derivatives tetration- 93
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 ... optimization functional-equations calculus-of-variations tetration- 17.9k
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 ... calculus sequences-and-series tetration- 93
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 ... exponentiation factorial tetration- 71
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 $e^{W(\log(x))}$, ... tetration- 11
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 ... complex-numbers tetration- 253
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 ... tetration- 19
Is there a proof showing super roots and super logarithms won't lead to a solution for the quintic?
So I am learning about tetrations and I just learned that tetrations are not elementary functions. When I heard that I remembered back to the statement that there is no general solution to the quintic ... tetration hyperoperation quintics- 209
Hyperoperators and zero-value conditions
Define for complex numbers: $f_0(z) = z+e; f_{n+1}(z+1) = f_n(f_{n+1}(z))$, where e = 2.1828... can be replaced with other bases. It's not hard to see that this is satisfied by: $f_1(z) = ez; f_2(z) = ... tetration- 161
Examples of closed forms of integrals with a power tower argument using W-Lambert function.
Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ... integration closed-form lambert-w tetration power-towers- 5,332
Why is $\sum_{n\ge1}\frac{\text B_\frac n2(-n,n)}{n^a b^n }=-\sum_{n\ge1}\frac{\left(\frac 2n-1\right)^n}{n^{a+1}b^n}$ close to (reciprocal) integers?
Here is a possible closed form of a sum with tetration in it made by equating coefficients of the Incomplete Beta function. This question is inspired by: Closed form of $$\sum\limits_{n=1}^\infty \... sequences-and-series approximation rational-numbers beta-function tetration- 5,332
Did I prove $\;{^{n}2} \equiv {^{n-1}2} $ $\,$ mod $n\;\;\;$ for $\;n \geq 2\;$?
Let's first state some properties of tetrations: They are one of the basic arithmetic functions, 4. hyperoperation to be exact. $\;{^{n}2}$ is in its basic, most elemental form. So we can't really ... elementary-number-theory discrete-mathematics solution-verification modular-arithmetic tetration- 39
Find all integers $n, n\gt2$ such that $n^{n-2}=x^n$ for some $x$
We can express this alternatively as $n^{n}=n^{2}x^{n}$. So the number raised to the power of itself has to be proportional to some number to the $n$-th power, but cannot be equal, naturally. I am not ... exponentiation tetration- 33
Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$
One can learn that for a power tower of height $n$: $$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$ giving a recursive relation. One might see that $n<-2$ cases are ... limits complex-numbers tetration power-towers experimental-mathematics- 5,332
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