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I want(ed) to analytically solve the transcendental equation $e^{\sin(x)} = \sin(e^x)$ for a closed form solution.

My working so far is:

$\frac{e^{ie^x}-e^{-ie^x}}{2i}=e^{\frac{e^{ix}-e^{-ix}}{2i}}$

$\frac{e^{2ie^x}-1}{2ie^{ie^x}}=e^{\frac{e^{2ix}-1}{2ie^{ix}}}$

$e^{2ie^x}-1 =2i(e^{ie^x+\frac{e^{2ix}-1}{2ie^{ix}}})$

$e^{2ie^x} = 2ie^{\frac{e^{2ix}-2e^{x + ix}-1}{2ie^{ix}}}+1$

$2ie^x = \ln(2ie^{\frac{e^{2ix}-2e^{x + ix}-1}{2ie^{ix}}}+1)$

I might've gotten a little bit further if I carried on but I couldn't find any summation-inside-logarithm theorems so I gave up - and I assume this is an unsolvable transcendental equation.

I know that most transcendental equations are unsolvable - barring lucky exceptions where terms can be canceled. And when I post these equations asking for questions on Quora, StackExchange etc. they immediately recognize the equation has no elementary solutions by just looking at it and post numerical approximations instead - is there a series of tests I can carry out on an equation to see if the transcendental equation has any elementary solutions? I'm not actually looking for approximations or practical takeaways, I'm just trying to practice my algebra skills for college.

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    Proofs that this or that equation has no elementary solution generally require very non-elementary mathematics, one generally relies on experience rather than tests to make the distinction. – Gerry Myerson Feb 09 '23 at 23:04
  • With great difficulty. – Mariano Suárez-Álvarez Feb 09 '23 at 23:05
  • @GerryMyerson Are you suggesting that trying to solve it, then getting stuck is the most time-efficient test? :) –  Feb 09 '23 at 23:08
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    I'm suggesting getting a feel for it is, in the long run, the best approach. There are papers about specific families of problems, e.g., papers about "integration in finite terms". – Gerry Myerson Feb 09 '23 at 23:13
  • Substituting $x=\ln(y)$ will simplify the first step of your work so far – Тyma Gaidash Feb 10 '23 at 00:24
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    @ashton, your equation seems not to have any solution for $x\leqslant0$, but it does have infinitely many solutions for $x>0$. Do you want to get all the solutions or just a few of them ? – Angelo Feb 10 '23 at 02:19
  • The exp-ln form of the equation can be transformed into an algebraic equation of more than one algebraically independent monomials. Therefore we cannot read an elementary inverse from the functions in the equation. But we don't know if an elementary inverse exists or not. – IV_ Feb 10 '23 at 17:01
  • Are you looking for tests for elementary solutions in general, solving $e^{\sin(x)}=\sin(e^x)$, a test on $e^{\sin(x)}=\sin(e^x)$ to see if it has elementary solutions, or something else? – Тyma Gaidash Feb 12 '23 at 00:52
  • General @TymaGaidash How could I determine if any transcendental equation eg. $e^{x^3+x^2+5x+12} = e^{3x}+e^{2x}+5e^x+12$ or $\cos(x) = x$ or $2^x=x^2$ etc. has elementary solutions or not with some sort of theorem. I heard somewhere that all transcendental equations are unsolvable for elementary solutions (except for Lambert W function sometimes) is that true? –  Feb 12 '23 at 01:18

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