sheldonison Wrote:The "upper/lower" properties of these two sexp solutions are very interesting, especially being able to convert one to the other. The "upper" solution approaches the larger fixed point at -infinity, and the lower solution approaches the smaller fixed point at +infinity.
Can this be applied to Kneser's fixed point solution for bases larger than (e^(1/e))? For base e, Kneser's solution, has complex values at the real number line,
No, Kneser though starts with the Schröder function at the primary (non-real) fixed point, which gives a superlogarithm/superexponential that has non-real values on the real axis. The superexponential is entire. However his aim is to have a real analytic solution. So he applies a conformal map to make it real on the real the axis, paying with the entireness.
Quote: and the function approaches the fixed point as x grows towards +infinity.Which function? The superexponential surely not. But superlogarithm/Abel function approaches the primary fixed point for \( z\to i\infty \).
Quote: But the desired solution has real values for all x>-2, and complex values for all x<-2 (except for the singularities). Moreover, the desired solution approaches the fixed point, as x approaches -infinity.
...
This has probably already been done, but can Kneser's base e solution, approaching a complex fixed point at +infinity, be converted it to another solution, approaching the fixed point at -infinity, with real values at the real number line, for all x>-2? Perhaps this line of reasoning isn't applicable because the solution wouldn't be holomorphic for all x>-2.
I think basically its already what Kneser did, however I dont know whether his solution approaches a fixed point for \( z\to -\infty \). This is also rather doubtful, because the development at the conjugate fixed point should bring the same solution. It can not converge to the non-real fixed point and to its conjugate at the same time.
Did you have a look at my introduction to Kneser's superlogarithm?