[rfc3092]

I believe this is correct: x/x = 1 everywhere except 0, where it has a removable singularity. So you can extend x/x holomorphically to full C.

This is completely different from the phenomenon described in the article: arccosh discontinuity can’t be dealt the same way. In fact complex analysis prefers to deal with it my making functions path-dependent (multi-valued).

[newobj]

PLEASE explain "So you can extend x/x holomorphically to full C" to someone with only a BSc in math/cs; something about this thread is giving me an existential crisis right now.

[rfc3092]

- function extension is defining a function where it is not defined

- <Adj> function extension is an extension that keeps (or gives) Adj property

- extended function is usually treated as originals if extension is good enough. Real analysis starts with defining real numbers and extending familiar functions onto them

- in this particular case we do not need C - even continuous extension on R works and agrees with x/x = 1 at 0

- holomorphic (analytic) extension makes function infinitely differentiable at every point of C

- because of the nature of discontinuity you can’t extend the simple arccosh in any reasonable way on C without introducing multivalued or path-dependent functions

- this continuity makes x/x=1 a reasonable simplification for CAS imo but not for complex functions as in the OP

- many things with point singularities in R have more structure in C, but x/x is not one of them. Even 1/x is of a different nature.

“You do not divide by zero” that forces you to carry x != 0 is more of a high-school construct than a real thing. Physicists ignore even more important stuff, and in the end their formulas work “just fine”.

[newobj]

Thank you, but, now I have 10 further “explain it to me” questions. (I never did analysis so this stuff is entirely over my head. I had one semester of algebraic structures. It was the hardest class I ever had in my life.)

[rfc3092]

As for existential crisis, you probably have missed this one: https://news.ycombinator.com/item?id=46962402

It was really fun