4
votes
1answer
237 views

Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration: $$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$ And now I am ...
8
votes
3answers
579 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?