Multiple exp^[1/2](z) by same sexp ?

[sheldonison]

....
We know from the previous posts that taking the half-iterate of a function close to exp makes a nice argument, but not a good example or counterexample.
( e.g. 2sinh^[1/2] )

2 Possibilities remain :

1) A power tower. Something like ln(x)^ln(ln(x))^... But thats even hard to prove being analytic.

2) sum z^n/(2^n!)
....

The Taylor series of half exponential's should behave similarly as real(z) increases since the exponential approximations behave similarly as real(z) increases. Can this be used to build an entire half iterate? What is the pattern?

The half exponential's I have worked most with are Kneser's tetration, which has singularities at -0.362+/-Pi i, plus a "quiet" singularity at L/L*, many others. But as z gets larger, it behaves very nicely. The other half exponential I've worked with is 2sinh^{1/2}(z), which has a singularity at 2i, as well as all of the iterates of 2i. I've also worked a little bit with the half exponential of exp(z)-1, and with sinh^{1/2}(z) both of which are generated from the parabolic case with a singularity at zero.

For exp(x) itself, if half is a Taylor series at x0, the recursive definition is \( \text{half}(\exp(x0)+x) = \exp(\text{half}(x0+\log(1+\frac{x}{\exp(x0)}))) \)

All three of these half iterates behave very nicely as real(z) gets larger, especially compared with tetration, and all three have similar Taylor series patterns, but I don't know what the pattern is. Here is the Taylor series for 2sinh^{1/2} at z0~=179.3912939581, whose half iterate is ~= e^^3, as well as the similar Kneser half iterate, and the half iterate of exp(z)-1. Can we quantify how much faster the derivatives decay, than exp(x+e^^2)? This might help in building an entire function, which approximates the half iterate as real(z) increases, or it may show why it may not be possible to generate an entire half exponential approximation; I will post more later; the equation above may prove very useful. All three of these half exponential functions have positive Taylor series coefficients up to around x^14 or x^15th.

Code:

2sinh^{1/2)(179.3912939581+x) ~=
       3814279.104760
+x^ 1*  116894.1861623
+x^ 2*  1545.365235620
+x^ 3*  11.55206923902
+x^ 4*  0.05384577308168
+x^ 5*  0.0001631177103376
+x^ 6*  0.0000003259720865567
+x^ 7*  0.000000000430018019731
+x^ 8*  3.722580437964 E-13
+x^ 9*  2.100799565010 E-16
+x^10*  7.704363816259 E-20
+x^11*  1.839961959607 E-23
+x^12*  2.873242189147 E-27
+x^13*  3.114122886329 E-31
+x^14*  7.902808213762 E-35
+x^15* -8.028657785155 E-37

KneserHalf(179.1155195732+x) ~=
       3814279.104760
+x^ 1*  117195.7788544
+x^ 2*  1553.468579782
+x^ 3*  11.64401876124
+x^ 4*  0.05442019863822
+x^ 5*  0.0001652812729209
+x^ 6*  0.0000003310497888663
+x^ 7*  0.0000000004374835751370
+x^ 8*  3.790743244215 E-13
+x^ 9*  2.138878310517 E-16
+x^10*  7.832144844649 E-20
+x^11*  1.865015582946 E-23
+x^12*  2.896225937469 E-27
+x^13*  3.309923535497 E-31
+x^14* -1.731956033634 E-35
+x^15* -3.215662487204 E-37

HalfExpz-1(178.0013796735+x) ~=
       3814279.104760
+x^ 1*  118320.9653987
+x^ 2*  1583.539131951
+x^ 3*  11.98264489348
+x^ 4*  0.05651283677712
+x^ 5*  0.0001730408420441
+x^ 6*  0.0000003488473243786
+x^ 7*  0.0000000004627633828955
+x^ 8*  4.009594804049 E-13
+x^ 9*  2.251097354280 E-16
+x^10*  8.155859990453 E-20
+x^11*  1.910315886523 E-23
+x^12*  2.897614845893 E-27
+x^13*  3.466010556218 E-31
+x^14* -1.279953070466 E-34
+x^15*  1.112984355720 E-37