Ivars Wrote:Do You find it not proper yet to single out the "mysterious" property of infinite tetration to turn certain real numbers x in x[4]n into complex as n->oo?
Dont know what about you are speaking. If you have two positive real numbers \( x \) and \( y \) then \( x^y \) is again a positive real number, hence the limit of \( x[4]n \),\( n\to\infty \) is a real number, if existing.
If you chose a negative number \( x \) then of course you may get a complex number as the result of exponentiation.
Quote:My speculative sentence would be related to speeds of growth:
Tetration is already so fast operation that it turns real numbers into complex;
You see thats already a property of triation=exponentiation which turns to real numbers into a complex one.
Quote:d(x[4]n)/dx= w[3]n ? Or will it be d(x[4]n)/dx= (w+lnx/x )[3]n?
No need to go into speculation, thats an easy exercise which you can solve for yourself:
\( \frac{dx[4]2}{dx}=\frac{dx^x}{dx}={x}^{x} \left( \ln \left( x \right) +1 \right) \)
\( \frac{dx[4]3}{dx}=\frac{dx^{x^x}}{d x}={x}^{{x}^{x}} \left( {x}^{x} \left( \ln \left( x \right) +1 \right)
\ln \left( x \right) +{\frac {{x}^{x}}{x}} \right) \)
\( \frac{dx[4]4}{dx}=\frac{dx^{x^{x^x}}}{dx}={x}^{{x}^{{x}^{x}}} \left( {x}^{{x}^{x}} \left( {x}^{x} \left( \ln
\left( x \right) +1 \right) \ln \left( x \right) +{\frac {{x}^{x}}{x
}} \right) \ln \left( x \right) +{\frac {{x}^{{x}^{x}}}{x}} \right) \)
obviously \( \frac{dx[4]n}{dx}\neq w^n \) and \( \frac{d x[4]n}{dx}\neq \left(w+\frac{\ln(x)}{x}\right)^n \) whatever \( w \) is. What is it btw?
