I have found an interesting and reliable method for obtaining a tetration result with a non-integer tetration exponent, the method of William Paulsen and Samuel Cowgill, which is based on limits and iterative functions. But here for the time being I am trying to find out about other, not so rigorous, ways to compute tetration, but which might give results close to those tetration results that are consistent with Paulsen and Cowgill's method, or close to the iterative (the most accurate) representation.
I wondered what definite integral could be used to describe the tetration so that the function passes through all points of the tetration results with integer tetration exponents. This question arose because it is currently known that the gamma function is a complete analytic extension (one of the variants) of the factorial, and that it was not previously known what the solutions were for the factorial, which would be taken from a non-integer number. Only then did we calculate the integral that realizes the factorial function as a full analytic continuation, where the curve clearly passes through all points of factorials of integers, starting from zero factorial. The factorial is an iterative function because it is based on recursion, so the gamma function is also an iterative function.
I wondered what definite integral could be used to describe the tetration so that the function passes through all points of the tetration results with integer tetration exponents. This question arose because it is currently known that the gamma function is a complete analytic extension (one of the variants) of the factorial, and that it was not previously known what the solutions were for the factorial, which would be taken from a non-integer number. Only then did we calculate the integral that realizes the factorial function as a full analytic continuation, where the curve clearly passes through all points of factorials of integers, starting from zero factorial. The factorial is an iterative function because it is based on recursion, so the gamma function is also an iterative function.