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base holomorphic tetration
#3
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?
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Messages In This Thread
base holomorphic tetration - by bo198214 - 11/05/2009, 02:12 PM
RE: base holomorphic tetration - by mike3 - 11/06/2009, 04:15 AM
RE: base holomorphic tetration - by mike3 - 11/06/2009, 11:58 AM
RE: base holomorphic tetration - by bo198214 - 11/06/2009, 12:12 PM
RE: base holomorphic tetration - by mike3 - 11/06/2009, 09:16 PM
RE: base holomorphic tetration - by bo198214 - 11/06/2009, 11:29 PM
RE: base holomorphic tetration - by mike3 - 11/07/2009, 12:23 AM
RE: base holomorphic tetration - by bo198214 - 11/07/2009, 08:17 AM
RE: base holomorphic tetration - by mike3 - 11/07/2009, 08:21 AM
RE: base holomorphic tetration - by bo198214 - 11/07/2009, 09:55 AM
RE: base holomorphic tetration - by bo198214 - 11/07/2009, 04:47 PM
RE: base holomorphic tetration - by bo198214 - 11/08/2009, 05:39 PM
RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:27 PM
RE: base holomorphic tetration - by mike3 - 11/08/2009, 08:25 PM
RE: base holomorphic tetration - by bo198214 - 11/08/2009, 08:44 PM
RE: base holomorphic tetration - by mike3 - 11/08/2009, 09:51 PM

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