Attempting to Define Tetration of Non-Integer Heights

Voiceover

Presenter(s) Information(s)

Morgan Holien

Abstract or Description

Tetration is a fundamental mathematical operation that follows in the sequence of addition, multiplication, and exponentiation. ⁿx is defined as x to the power of itself, n times. Naturally, the question arises: What happens if n is a non-integer? Many generalizations to non-integers have been produced over the history of tetration, all of which fall short in one way or another. As such, there is no universally accepted definition of non-integer tetration. A viable non-integer extension is crucial to the physical application of tetration, as the real world so rarely involves pure integers.

To address this shortcoming, I am attempting to define an extension for non-integer heights that meets all ideal extension requirements. Using the known functional square root of ln(x+1), I was able to show that ^-3/2(e^1/e) ≈ -1.295. Using basic identities, I then calculated e^(1/e) tetrated to other heights, which are consistent with the values produced by other non-explicit methods. Because this derivation was discovered quite recently, I have yet to fully generalize the process, though I have many ideas on how to do so. Additionally, I discovered several new tetrational identities, which may be used to further characterize the behavior of tetration.

With the framework laid, it is only a matter of time before I can derive a complete extension for non-integer tetration. This may ultimately allow the operation to be fully integrated into the study of the natural world.