10/30/2016, 11:02 PM
(This post was last modified: 10/30/2016, 11:08 PM by Vladimir Reshetnikov.)

After some study of different approaches to an extension of tetration to fractional or complex heights, and many numeric experiments, I came to the following conjecture, that I am currently trying to prove:

Let be a fixed real number in the interval . There is a unique function of a complex variable , defined on the complex half-plane , and satisfying all of the following conditions:

* .

* The identity holds for all complex in its domain (together with the first condition, it implies that for all ).

* For real is a continuous real-valued function, and its derivative is a completely monotone function (this condition alone implies that the function is real-analytic for ).

* The function is holomorphic on its domain.

Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it.

Let be a fixed real number in the interval . There is a unique function of a complex variable , defined on the complex half-plane , and satisfying all of the following conditions:

* .

* The identity holds for all complex in its domain (together with the first condition, it implies that for all ).

* For real is a continuous real-valued function, and its derivative is a completely monotone function (this condition alone implies that the function is real-analytic for ).

* The function is holomorphic on its domain.

Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it.