4 ms·

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

• rfc3092 7mo ago
  - 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 7mo ago
    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 7mo ago
  As for existential crisis, you probably have missed this one: https://news.ycombinator.com/item?id=46962402 https://news.ycombinator.com/item?id=46962402 It was really fun