11/17/2008, 04:11 AM

sheldonison Wrote:...

To me, it would seem as if the first couple of derivatives at x=-1,0 were the same for both solutions, than it is likely that they're the same solution, though the two could still diverge in the higher derivatives.

I had submitted the evaluation of the first derivative with 14 (I hope) decimal digits.

In the similar way, the second derivative can be evaluated, and so on.

But I like the approach by Bo: it is better to prove, than to compare the numerical evaluations. We already have proof that no other tetration is allowed to be holomorphic in the complex plane with cutted out set .

By the way, how about to add some small amount of finction

?THis function has all the derivatives along the real axis, and all of them are zero at points 0 and -1.

Consider the modification of tetration to

with corresponding extension from the range [-1,0] usind the recurrent equation. At the real axis, the modified tetration tem has the same "expansion" as the tetration in these points; however, such a modification destroys the holomorphism.