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Comment by rfc3092

6 hours ago

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).

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rfc3092

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newobj  5 hours ago

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  2 hours 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”.