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Complex numbers do connect trigonometric functions with hyperbolic functions and exponents in closed form. Has anybody ever proposed an algebraic system that would connect in a similar way trigonometric functions and inverse trigonometric functions (logarithms)?

I am asking this because taking roots for instance can be expressed through exponentiation while logarithms are standing quite apart. If logarithms can be expressed via exponentiation, then all elementary functions can be expressed via addition, multiplication and power.

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  • $\begingroup$ Does $\lim_{a\to 0} \frac{x^a-1}{a} = \log(x)$ count? $\endgroup$
    – Suvrit
    Commented Nov 13, 2016 at 4:25
  • $\begingroup$ @Suvrit interesting hit, actually! I think it does not count per se, but can be a potential basis for contruction of a numerical system (for instance, adding such $\epsilon$ that $\frac{x^\epsilon-1}{\epsilon}=\log x)$. $\endgroup$
    – Anixx
    Commented Nov 13, 2016 at 4:31
  • $\begingroup$ Maybe it would help to try and state more precisely what "connect" should mean? $\endgroup$
    – Nate Eldredge
    Commented Nov 13, 2016 at 4:48
  • $\begingroup$ @Nate Eldredge By "connect" I meant "express in closed form". $\endgroup$
    – Anixx
    Commented Nov 13, 2016 at 5:04
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    $\begingroup$ Looks like a duplicate of mathoverflow.net/questions/237919/… (if you agree, please upvote this comment so that it appears "above the fold") $\endgroup$
    – Timothy Chow
    Commented Nov 13, 2016 at 21:38

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