Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ?

[tommy1729]

Im very intrested in the behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) for real x > 0.

Lists and plots are welcome !

Regards

tommy1729

[tommy1729]

One of the reasons im intrested in this is because these functions must be between id(x) and x^2 or id(x) and sqrt(x) resp.

So no silly slow or silly fast growth rates. Also its an analogue to the question what lies "between" polynomial and exponential ?

This question appears to be what lies "between" linear and squared ?

Many functions grow like ...(exp(x)^a)(x^b)(ln(x)^c)...* ...(sexp(x)^a_2)(slog(x)^b_2).... and this one might be different. And even if similar that is also intresting imho.

Since these functions grow at normal rates we might be able to do 'normal' math such as calculus or number theory(*) or simpler recursions.(* by using rounding )

[tommy1729]

Am I the only one who finds this intresting ?

[cosurgi]

I'm not sure if I get your formula right......

This is what I type in gnuplot:

plot exp(2*(log(x))**0.5)**0.5
replot exp(0.5*(log(x))**0.5)**0.5

The "**" symbol means raising to power, in gnuplot.

[Balarka Sen]

@cosurgi : by exp^[1/2](x) and log^[1/2](x), he means half-exponential and half-logarithms (logarithms iterated 1/2 times) not raised to the power of 1/2.

[cosurgi]

@cosurgi : by exp^[1/2](x) and log^[1/2](x), he means half-exponential and half-logarithms (logarithms iterated 1/2 times) not raised to the power of 1/2.

oh, ok. So I don't know how to plot that

EDIT: unless you can give me some formula for half-iterated logarithm and exp.

[Balarka Sen]

unless you can give me some formula for half-iterated logarithm and exp.

They are both defined as f(f(x)) = e^x and f(f(x)) = log(x), respectively. A closed form expression can be obtained for the former one in terms of natural tetration (base-e tetration) if I am not wrong but I don't know anything about the later one, unfortunately.

[tommy1729]

Btw it is easy to show that exp^[1/2](2 log^[1/2](x)) grows slower than any power law.