Rational operators (a {t} b); a,b > e solved

[bo198214]

Yes it would be nice to have it analytic on the real axis, I think it should be potentially analytic for at least (-oo, 1).

Yes.

No I haven't compared the derivations, I'll do that though. It's hard for me to think up tests I can do with only highschool math under my belt

You do very well

but I think that for the same reason the gamma function is written \( \Gamma(n+1) \) instead of \( n! \)

Oh dear thats another topic! This definition of the Gamma function (being off by one from factorial) is imho a history misdevelopment. There is no use at all of it, and it confuses the minds.

, logarithmic semi-operators should be written using their own notation--just to be clear this is only one extension of hyper operators. I know that simply stating this is the natural extension of hyper operators will step on a lot of people's toes.

Sure. I mean even for tetration there are different ways of extension. And long fights about which is the best/the most natural etc....