Is There an Operation Between Exponentiation and Tetration?
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The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as
begin{align*}
text{exp} (a, n) = a^n=acdot acdots a
end{align*}
and tetration as
begin{align*}
text{tet}(a, n)&=text{exp} (text{exp}(cdots text{exp}(a, n)))\
&=a^{a^{cdots^{a} } }
end{align*}
where $a$ and $n$ are integers, then is there a type of operation that falls between these two? In other words, is there a function $f(n)$ such that $text{exp} (a, n) < f(n) < text{tet}(n, a) $ for (most) integers $n$? My guess would be the factorial function but I'm not sure.
tetration
asked Dec 3 at 12:43
Alex S
30517
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up vote
0
down vote
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The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as
begin{align*}
text{exp} (a, n) = a^n=acdot acdots a
end{align*}
and tetration as
begin{align*}
text{tet}(a, n)&=text{exp} (text{exp}(cdots text{exp}(a, n)))\
&=a^{a^{cdots^{a} } }
end{align*}
where $a$ and $n$ are integers, then is there a type of operation that falls between these two? In other words, is there a function $f(n)$ such that $text{exp} (a, n) < f(n) < text{tet}(n, a) $ for (most) integers $n$? My guess would be the factorial function but I'm not sure.
tetration
asked Dec 3 at 12:43
Alex S
30517
Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as
begin{align*}
text{exp} (a, n) = a^n=acdot acdots a
end{align*}
and tetration as
begin{align*}
text{tet}(a, n)&=text{exp} (text{exp}(cdots text{exp}(a, n)))\
&=a^{a^{cdots^{a} } }
end{align*}
where $a$ and $n$ are integers, then is there a type of operation that falls between these two? In other words, is there a function $f(n)$ such that $text{exp} (a, n) < f(n) < text{tet}(n, a) $ for (most) integers $n$? My guess would be the factorial function but I'm not sure.
tetration
asked Dec 3 at 12:43
Alex S
30517
The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as
begin{align*}
text{exp} (a, n) = a^n=acdot acdots a
end{align*}
and tetration as
begin{align*}
text{tet}(a, n)&=text{exp} (text{exp}(cdots text{exp}(a, n)))\
&=a^{a^{cdots^{a} } }
end{align*}
where $a$ and $n$ are integers, then is there a type of operation that falls between these two? In other words, is there a function $f(n)$ such that $text{exp} (a, n) < f(n) < text{tet}(n, a) $ for (most) integers $n$? My guess would be the factorial function but I'm not sure.
tetration
tetration
asked Dec 3 at 12:43
Alex S
30517
asked Dec 3 at 12:43
Alex S
30517
asked Dec 3 at 12:43
Alex S
30517
asked Dec 3 at 12:43
Alex S
30517
asked Dec 3 at 12:43
Alex S
30517
30517
Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49
add a comment |
Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49
Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49
Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49
add a comment |
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Yes a factorial function is between. Because tetration has a much higher divergence rate than exponentiation, there are lots of functions in between, including factorials, double exponentials, any function like $operatorname{tet}(n,b)$ for fixed $b$.
– Dark Malthorp
Dec 3 at 12:49