4
votes
1answer
84 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
1
vote
5answers
191 views

Limit of a recursively defined bivariate function.

Let m and n be positive integers. Let $f(m,0)=m$ Let $f(m,n)= e \ln(f(m,n-1))$ $$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$ Where $C$ is a constant. It seems ...
4
votes
1answer
305 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?