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

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Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - tommy1729 - 02/20/2013

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


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - tommy1729 - 02/20/2013

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 )


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - tommy1729 - 02/21/2013

Am I the only one who finds this intresting ?


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - cosurgi - 02/23/2013

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.




RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - Balarka Sen - 02/24/2013

@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.


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - cosurgi - 02/24/2013

(02/24/2013, 06:54 AM)Balarka Sen Wrote: @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 Smile

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


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - Balarka Sen - 02/24/2013

cosurgi Wrote: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.


RE: Behaviour of exp^[1/2](2 log^[1/2](x)) and exp^[1/2](1/2 log^[1/2](x)) ? - tommy1729 - 02/26/2013

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