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The upper superexponential
#11
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 .

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 . 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?
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Messages In This Thread
The upper superexponential - by bo198214 - 03/29/2009, 11:23 AM
RE: The upper superexponential - by andydude - 03/31/2009, 04:29 AM
RE: The upper superexponential - by sheldonison - 04/03/2009, 03:06 PM
RE: The upper superexponential - by bo198214 - 04/03/2009, 04:22 PM
RE: The upper superexponential - by sheldonison - 04/05/2009, 12:45 PM
RE: The upper superexponential - by bo198214 - 04/06/2009, 06:35 AM
RE: The upper superexponential - by Kouznetsov - 05/10/2009, 02:13 PM
RE: The upper superexponential - by sheldonison - 05/11/2009, 12:55 PM
RE: The upper superexponential - by bo198214 - 05/11/2009, 01:21 PM
RE: The upper superexponential - by sheldonison - 05/11/2009, 08:12 PM
RE: The upper superexponential - by bo198214 - 05/11/2009, 08:31 PM
RE: The upper superexponential - by Kouznetsov - 05/12/2009, 08:54 AM
RE: The upper superexponential - by bo198214 - 06/01/2009, 07:24 PM
RE: The upper superexponential - by tommy1729 - 04/05/2009, 07:05 PM
RE: The upper superexponential - by sheldonison - 04/22/2009, 05:02 PM
RE: The upper superexponential - by bo198214 - 04/22/2009, 05:34 PM

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