# Tagged Questions

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

46 views

### pentation of uncountable ordinal

$\omega↑↑↑\omega = \Gamma_0$ Feferman's ordinal collapsing function: $\ θ(Г_Ω) = θ(Ω_2)$ $\ θ(Г_{Ω_2}) = θ(Ω_3)$ ...etc. This means: $θ(Ω_2) = θ(Ω↑↑↑Ω)$ $θ(Ω_3) = θ(Ω_2↑↑↑Ω_2)$ ...etc. ...
85 views

### Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
67 views

### How do hyperoperations like tetration exist if operations are seperate relations and not repeatitions of each other.

I've run into a bit of a conflict in my fundamental understanding of concepts in math. I've always known the arithmetic operation to be extensions of each other. Multiplication is repeated addition, ...
232 views

### Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
27 views

### What is the generating function $G(x,y)$ for powertowers of $x$?

I'm looking for a generating function $G(x,y)$ for powertowers of $x$ such that $G(x,y) = 1 + xy + x^x {y^2 \over 2!} + x^{x^x} {y^3 \over 3!} + x^{x^{x^x}} {y^4 \over 4!} + ...$ Is there any ...
43 views

### Is $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)) = H_n(\displaystyle\lim_{h \to 0} f(h), \displaystyle\lim_{h \to 0} g(h))$ true for all $n$?

Consider the limit $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)),$ where $H_n(a, b)$ denotes the $n$th hyperoperation $H_n(a,b) = a \uparrow^{n-2}b$ with both $f(x)$ and $g(x)$ being continuous and ...
41 views

336 views

### Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
145 views

183 views

118 views

### Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
438 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?
I here define the hierarchy of basic mathematical operators and their respective "inverse" operation (see hyperoperation).  \begin{array}{c|c|c|} & \text{Operator} & \text{"Inverse"} \\ \...