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The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.

Is there possible an extension of real/complex numbers in which logarithms and inverse trigonometric functions can be expressed in terms of exponentials/trigonometric functions and vice versa in closed form?

P.S. I have asked here but is seems people there just do not understand the question.

What I am talking about is something like this: $$\frac1\pi\ln \left(\frac{w-\frac{z}{\pi }}{w-1+\frac{z}{\pi }}\right)=\frac1z\cos (2wz)$$

or this:

$$\ln(\sin(w + z)) = 1/z \cos(2 w z)$$

Where $w$ is some element of the extended field, not a complex number. Is this possible?

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    $\begingroup$ Anything is possible: take the set of solutions $z,w$ to your equation, as a Riemann surface inside $\mathbb{C}^2$. Then on that set, $w$ is a function satisfying your equation, by definition. The field of meromorphic functions of the $z$ variable pulls back, from projection $(z,w) \mapsto z$ to live inside the meromorphic functions on that Riemann surface. Not a satisfying or explicit solution, more like a solution by definition. $\endgroup$
    – Ben McKay
    Commented May 3, 2016 at 10:38
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    $\begingroup$ I don't know. What is a numerical field? It won't be a number field, or an algebraic extension field. $\endgroup$
    – Ben McKay
    Commented May 3, 2016 at 10:48
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    $\begingroup$ It is a field: commutative, associative, distributive, and every nonzero element has a multiplicative inverse. $\endgroup$
    – Ben McKay
    Commented May 3, 2016 at 11:08
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    $\begingroup$ "people there just do not understand the question". Well, I don't mind confessing my thickness, but I don't understand the question either. I don't know what a "numerical field" is, and I don't know how you're going to extend the exponential/logarithm to that set. Except tautological answers like BenMcKay's, I just don't know what new "numbers" you'd expect. Just because you state "I wish I knew some system in which such and such formula is true" does not mean you ask a well-formed question and that everybody is supposed to understand its meaning. $\endgroup$
    – Loïc Teyssier
    Commented May 3, 2016 at 11:50
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    $\begingroup$ I think the question is whether the log function can be expressed using a finite number of compositions of arithmetic operations and the exponential function, starting with a finite number of constants ($w$ in Anixx's notation). The answer is "obviously" no but I think it is not trivial to prove it. $\endgroup$
    – Timothy Chow
    Commented Nov 13, 2016 at 21:47

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As you suggested at a related site, the natural log can be expressed via the shadow of a hyperfinite partial sum of the harmonic series.

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  • $\begingroup$ What you are referring to? What site?.. $\endgroup$
    – Anixx
    Commented May 8, 2016 at 7:41
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    $\begingroup$ I meant at this site at a different question. $\endgroup$
    – Mikhail Katz
    Commented May 8, 2016 at 7:43
  • $\begingroup$ Oh I see. But this is still not closed form, is it? $\endgroup$
    – Anixx
    Commented May 8, 2016 at 7:44

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