elementary superfunctions

[bo198214]

Thatswhy I always try to find for elementary solutions whether they are regular at some fixed point because this reduces the number of real analytic solutions to two at one fixed point (analogously to \( \exp_{\sqrt{2}} \)) up to x-translation.

I want to illustrate this phenomenon with a picture of the two regular super-functions \( F_{1,+}(x)=\cosh(2^x) \) and \( F_{1,-}(x)=\cos(2^x) \) of \( f(x)=2*x^2-1 \) at the fixed point 1.
   
The upper curve is \( \cosh(2^x) \) and the lower curve is \( \cos(2^x) \).
We see that they have both same asymptote to the left, which is the fixed point 1.

Compare this with the both super-exponentials at 4 (these are \( F_{4,5} \) and \( F_{4,3} \)) in the picture in this post.

This is a general behaviour of the two real regular super-functions at one fixed point: Either to the left or to the right (depending whether the derivative at the fixed point is bigger or smaller than 1) they both approach the fixed point, one from above the other from below.