I've also been toying with this, too. It appears, however, that it continues to

real values, not complex values, for

.

Consider the following “regular iteration” limit formula:

http://math.eretrandre.org/tetrationforu...10#pid1610
where

is the attracting fixed point, i.e.

.

There's also a series formula, apparently (with respect to "w" in

, so setting w = 1 yields the tetrational), but I haven't yet figured out how one is supposed to evaluate the general coefficient. How can that be done?

To test the analytic continuation, we can use the Cauchy integral: if

is analytic, then we derive the powerseries coefficients at

via

and

.

This seems to provide a more efficient algorithm for the recovery of the coefficients, than straight numerical differentiation from the difference quotient (which seems to require more rapidly-escalating levels of numerical precision).

We can now choose some

close to

, set

for some fractional tower

where

is obtained from the regular formula, and a path that encircles it, but does not leave the kidneybean ("Shell-Thron" region) of convergence, e.g. a small circle round the point. Then, by increasing n, we obtain the Taylor coefficients. For

, expanded about

, using a circle of radius 0.01, we get the following estimates for the first 25 coefficients:

Code:

`a_0 ~ 1.24622003310`

a_1 ~ 0.447921100148

a_2 ~ -0.194428566238

a_3 ~ 0.143167873861

a_4 ~ -0.144967399774

a_5 ~ 0.182159301224

a_6 ~ -0.263407426426

a_7 ~ 0.417098477762

a_8 ~ -0.702204147888

a_9 ~ 1.23535078363

a_10 ~ -2.24705897139

a_11 ~ 4.19678269601

a_12 ~ -8.00913024091

a_13 ~ 15.5621057277

a_14 ~ -30.7029302261

a_15 ~ 61.3746543453

a_16 ~ -124.093757768

a_17 ~ 253.427621734

a_18 ~ -522.152429631

a_19 ~ 1084.31773542

a_20 ~ -2267.61147731

a_21 ~ 4772.09902388

a_22 ~ -10098.4528601

a_23 ~ 21464.8938685

a_24 ~ -45852.6753827

For

, we can use this get

, which is real, not complex. How does that agree with other methods of tetration for bases greater than

? This series should have radius of convergence 0.42, determined by the distance to the nearest singularity/branchpoint, which is at z = 1.

I'm not sure of a formal proof of the "continuability", though one approach may be to try and differentiate the regular iteration formula, then prove that the limit of the derivative as

converges -- in order for it to switch to non-real complex values as

is passed, that point would have to be some sort of singularity, like a branch point, and so the function would not be differentiable there, and if it is, then that is not the case.

I'll see if maybe I can get some graphs on the complex plane but calculating the regular iteration is a bear as it requires lots of numerical precision, at least for the limit formula. Maybe that series formula would be better?